A guide for grade 10 quadratics, including 3 forms and more!
Table of Contents
- What is quadratics?
- Parabolas & parts
- Quadratic vs Linear
- Vertex form parts
- Graphing from vertex form (+ step pattern)
- Finding the vertex given the equation
- Factored Form + roots
- Finding the vertex with the roots
- Graphing factored form
- Standard Form
- Standard form graphing
- Multiplying binomials + special products
- Algebra tiles
- Factoring (common, grouping, simple, complex, perfect square trinomial, difference of squares)
- Completing the square
- Solving quadratic equations (quadratic formula, factoring, isolating x)
- Word problems (motion, revenue, area, consecutive numbers, right triangle, optimization)
- Reflection + an assessment
What are quadratics?
PARTS OF A PARABOLA
The Vertex: The Main Character
Optimal or Minimum/Maximum value
X-Intercept, Zeros, and Roots
Some Other Interesting Facts About A Parabola
- A parabola can open up or down. This will be shown in a section below.
- The vertex is the point where the axis of symmetry and the parabola touch. This is the optimal value (minimum or maximum).
QUADRATIC EQUATION VS LINEAR EQUATION
What's the difference?
- A linear equation creates a straight line, whereas a quadratic equation creates a curved line.
- In a quadratic equation, the variable is squared.
- In a linear equation, the first differences are constant. In a quadratic equation, the second differences are constant.
"Wait, wait, wait! Did you just say second differences?"
Yes, I did. In grade 9, we worked with first differences, the differences between two consecutive terms. The second differences are the differences between the first differences. Confusing? Here's a picture to help:
What Is Vertex Form
An example of vertex form is:
The a value can change a few things in the graph:
- If a is greater than 0 (meaning positive), then the graph opens upwards. When a is positive and the parabola opens upwards, the vertex is a minimum value.
- If a is less than 0 (meaning negative), then the graph opens downwards. When a is negative and the parabola opens downwards, the vertex is a maximum value
- When a is negative, it is called a vertical reflection, or reflection in the x-axis. This is because the graph seems to have reflected and is now facing downwards.
- If a is greater than 1, then the graph is stretched. This makes the graph narrower.
- If a is greater than 0, but less than 1 (meaning it's a decimal/fraction), then the graph is compressed. This makes the graph wider.
- The h value moves the vertex left or right, and this is called a horizontal translation.
- The translation always starts at (0,0).
The vertex is moved in the opposite way of the sign.
- This means that is h is negative, then the vertex will move h units right (which is the positive part of the x-axis).
- If h is positive, then the vertex will move h units left (which is the negative part of the x-axis).
- The h is also the x-coordinate of the vertex, but with the opposite sign.
- The k value moves the vertex up or down, which is called a vertical translation.
- Like the h value, the vertical translation should start from (0,0). You can put the horizontal and vertical translation together to find out where you must move from (0,0).
Unlike h, the k value moves in the same way as the sign, meaning:
- If k is positive, then the graph will move k units up.
- If k is negative, then the graph will move k units down.
- The k is the y-coordinate of the vertex.
Put them together!
First, let's try out an example, and see what information we have. We'll use the equation:
Lets look at the a value:
The a value is -2.
- We remember that a changes stretch/compression and reflection.
- Because a is negative, the parabola is vertically reflected. This means that the parabola opens down.
- Also, we can see the 2 in a. This means that the parabola is stretched by a factor of 2.
Now you might be wondering what stretched is, and how we use it. That will be explained later, when we learn to graph the vertex form.
Next we can look at the h value:
The h is -1.
- We remember that h is responsible for the horizontal translation. We also remember that the vertex will translate the opposite direction of the sign.
- You can see that the h is negative. This means that it will translate to the right, and it will translate by 1 unit.
After that, we can look at the k.
The k is 3.
- The k is used to translate the graph vertically. This will move the graph up or down.
- Remember, unlike the h, the k moves in the same direction of the sign.
- Because the k is +3, the graph will move 3 units up.
Finally, we know what will happen to our graph. It is reflected vertically, stretched by a factor of 2, translated 1 unit right, and translated 3 units up.
- Notice the last two points: translated 1 unit right, and translated 3 units up.
- Earlier, when describing h and k, we said that the translation would start at (0,0).
- If it moves 1 to the right (x-axis), and 3 up (y-axis), what would the coordinates of the vertex be in (x,y)?
The vertex would be (1,3). This is why this form is called vertex form, because the vertex is written in the equation. Look at y=-2(x-1)^2+3.
- See -1? That's the 1 of the vertex (x-axis). Remember, the sign is opposite.
- See +3? That's the 3 of the vertex (y-axis).
Try to make the equations for the following:
- Stretched by a factor of 2. Vertex: (-3,4)
- Reflected in the x-axis. Compressed by a factor of 0.5. Vertex: (2,-5)
- Reflected in the x-axis. Stretched by a factor of 4. Vertex: (1,1)
Write the properties for the following parabolas based on the equations. Include stretch/compression, whether or not it is reflected in the x-axis, the vertex along with the translations it went through to get there, and if the vertex is a minimum or maximum point.
We are almost ready to graph using vertex form. All that is left is to learn of something called the step pattern:
The Step Pattern
Take a look at the graph for y=x^2:
This is what is meant by the variable is squared. The y is equal to x times x. So if x is 4, the y is 16. This is a step pattern. The step pattern is a way to describe what the y is when the x is a certain number, and in this case: 1 and 2. Remember, the step pattern for this parabola is "Over 1, Up 1. Over 2, Up 4".
You begin the step pattern at the vertex. You can use this step pattern moving both left and right of the vertex, because a parabola is symmetrical from the axis of symmetry. This means that you can go left 1, up 1, left 2, up 4 AND right 1, up 1, right 2, up 4. You use the step pattern to plot points on your graph so you can sketch the parabola.
Now you might be wondering "Okay, so I know how to graph y=x^2. Just start at (0,0), and go over 1, up 1, over 2, up 4. But how would I graph something like y=-2(x-1)^2+3?"
The step pattern will change as the a value changes. When the graph is stretched or compressed, the steps will increase or decrease, accordingly. Also, if a is negative, then the step pattern will be "Over, Down. Over, Down" instead of up.
To find a step pattern for an equation, you can use the basic step pattern of Over 1, Up 1, Over 2, Up 4. We can just multiply all of the y values by the a value. For our equation y=-2(x-1)^2+3, we will multiply the y values by -2. It will look like:
Over 1, up 1a. Over 2, up 4a.
Using this, the new pattern would be: Over 1, down 2. Over 2, down 8.
Do you notice how this changes? When the a value changes, the step pattern changes. To recap, just multiply the y value by the a value, and you found the new step pattern.
So now we have the vertex (because of the translations), the direction of opening (reflection), and the step pattern. we are ready to graph!
Graphing With Vertex Form
Let's look at y=-2(x-1)^2+3, and how to graph it. Here are the steps:
1. Notice the stretch and the direction of opening.
In this equation, the vertical stretch is by a factor of 2. Also, the direction of opening is down because it is negative.
2. Find and plot the vertex.
We have already found the vertex for this equation. By looking at the equation, we know that the vertex is (1,3). We can plot this point on the graph.
3. Find points on the graph.
You can do this by finding the step pattern of the graph. Remember, to find the step pattern by an equation, multiply the original step pattern by the a value. It is Over 1, up 1a, Over 2, up 4a. The new step pattern for this equation is "Over 1, down 2, Over 2, down 8".
Begin at the vertex that you plotted, and use the step pattern to find 2 points on both the right and left of the vertex. This means go left 1, down 2, left 2, down 8. Also go right 1, down 2, right 2, down 8.
Another way to do this is to sub in a point on the x-axis into the equation as the x, and solve for y. The x would be the x-coordinate of this point, and the y that you solved for would be the y-coordinate. For example, if our vertex is (1,3), I can look for the y when x is 2. so it would be:
So a point on my graph would be (2,1). However, you should probably use the step pattern, as it is faster and you wont need to do the equation to find every point.
4. Sketch the parabola.
Now that you have the vertex and 4 other points, you can draw the U-shaped parabola. Just sketch it like its connect the dots, but extend it a bit and add arrows to each side. Your graph should look something like:
Practice Word Problem
1. Graph this equation.
2. For how long was the ball in the air?
3. What was the highest point the ball reached, in meters?
4. At how many seconds did the ball reach this height?
Because this is a practice problem, we will solve this for you, so you can take this as an example.
1. Graph this equation.
For this question, we can look at the 2 x-intercepts. The first one on the left would represent where the ball was kick. From there to the other x-intercept would be the time that the ball was in the air.
The ball was in the air for 12 seconds.
3. What was the highest point the ball reached, in meters?
Now we need to look at the vertex, as it is the maximum point in this graph. We can look for the y-coordinate of the vertex, and that will be our highest point.
The highest the ball reached was 9 meters.
4. At how many seconds did the ball reach this point?
For this, we can look at the x-coordinate of the vertex.
The ball reached 9 meters at 6 seconds.
Determining the Equation of a Parabola When Given the Vertex
Let's say the vertex is (-2,8), and the parabola passes through the point (2,0).
Remember how we found the vertex by looking at h and k? Also, remember how it was mentioned that the vertex was actually in the equation and that's why the form was called vertex form. Well, if you're given the vertex in a question, you already have 2 variables: h and k.
h is the x-coordinate of the vertex, but with the opposite sign. In this case, it would be 2.
k is the y-coordinate of the vertex. In this case, it would be 8
You can put these points into the vertex form equation. So at this point, our equation is:
Next, we know that this parabola passes through the point (2,0). We can substitute these values in for x and y in our equation. Now, our equation is:
At this point, the only variable left is a, meaning that we can solve for a by isolating it.
Now that we have a, we can place it into our equation, and change x and y back to variables. Finally, our equation is:
Remember, when you're solving one of these questions, first, put the h and k values into the equation, then substitute x and y to solve for an a value, and finally, but a into the equation to finish the equation.
Let's try another one.
"Find the equation for a parabola that was a vertex of (2,6) and passes through the point (5,3)."
The solution would look like:
Therefore, the equation for the parabola is:
What is Factored Form?
Finding the Vertex with x-intercepts
Graphing with the x-intercepts and the Vertex
1. Plot the x-intercepts. We will put points at (5,0) and (-3,0). Remember, for x-intercepts, the y-value is 0.
Let's try another example:
What is standard form?
Graphing standard form
Factoring will be taught below, after multiplying binomials. However, here is an example of graphing standard form. Keep in mind what is happening so that you will understand how to graph.
What is a binomial?
If you're wondering how to convert standard form to factored form, that is done by factoring, which is further down on this site.
Perfect Square Binomials
These special binomials are perfect squares, meaning all of the binomial is squared. This is also what you would use to convert vertex form to standard form. Examples are shown below:
Product of sum and difference
What is factoring?
- Common factoring
- Factoring by grouping
- Factoring simple trinomials
- Factoring complex trinomials
- Special products
- Factoring perfect square trinomials
Now, let's try a harder example:
Factoring by grouping
Factoring simple trinomials
- The b value of standard form is equal to r + s of factored form.
- The c value of standard form is equal to r x s of factored form.
To do this, we need to find factors of c, and whichever pair equals to b is the factors we are looking for. Lets look back at our example.
So now we have the numbers that have a product of 2 and a sum of 3. We can finally fully factor.
Since we already know that the factors of x^2 are x and x, we can place those.
Factoring complex trinomials
- First look for common factors, and remove them. If possible, turn the complex trinomial into a simple trinomial by factoring the a value from the trinomial. If you can do this, then factor it like a simple trinomial.
- Write the factors of the first and last term. In a simple trinomial, you only need the factors of the 3rd term, because the factors of the first term were always x and x. However, in a complex trinomial, the a value is not 1, so you must find the factors.
Once again, you will need specific factors. Each of the factors of the first term must multiply with one of the factors of the third term. Then, when you add it all up, it should equal the middle term.
Factoring the Difference of Squares
Look back to MULTIPLYING BINOMIALS, and look at The Product of Sum and Difference. Remember:
This expression follows the the a squares subtract b squared expression. In this case, x is a and 100 is b. This must mean that it is also equal to (a+b)(a-b). If we want to factor this expression, we must get it back to (a+b)(a-b).
Another reason why this equation must be (a+b)(a-b) is because there are 2 squares, and the sign is negative. A sign becomes negative when it is positive and negative, meaning that one must be positive and the other must be negative.
This means a is 5x and b is 8. We can fill this into (a+b)(a-b).
Factoring Perfect Square Tinomials
To factor this, you can just take a and b, put it in a bracket and square it. The factored form of this looks like:
COMPLETING THE SQUARE
Put it together
1. Make the standard form equation look similar to vertex form.
This can be done by factoring our the a value and separating the c to make it look like k from vertex form.
2. Expand what's in the brackets to make it a perfect square trinomial.
You will need to add and subtract the c from the perfect square trinomial.
3. Factor the perfect square trinomial.
4. Apply the distributive property from a.
5. Add like terms.
This will be better shown with an example:
In this step, we will factor our the a value, and separate the c by brackets.
In this step, we need to turn what is in the middle into a perfect square trinomial. To do so, we need a c value. In order to get the c value we need for a perfect square trinomial, divide the b value by 2. (This is because in a perfect square trinomial, the middle value equals to 2ab from the binomial.) In this case, -4 divided by 2 is -2. -2^2 is our c value, because a perfect square trinomial is a^2 + 2ab + b^2.
We will need to add the c value, and then subtract it. This is because we cannot just add a number into the equation, as that changes the equation completely. However, + c - c = 0, so we can do that.
In the final part of this step, the perfect square trinomial is in a bracket in a bracket. This is only to help organize, and is not necessary. Everything inside the bracket will be multiplied by the distributive property of 3 soon, however, putting the perfect square trinomial in a bracket just helps you see what will be factored.
Now we will factor the perfect square trinomial. We need to find the square roots of x^2 and 4, which is x and 2. However, the b term is negative, meaning that the 2 is negative. Our factored form of the perfect square trinomial is (x-2)^. We can keep this inside of the bracket, as it will still be multiplied by 3.
The -4 (which was used to counter the +4 from the perfect square trinomial) will remain inside the bracket, and wasn't factored because it wasn't a part of the perfect square trinomial. It will also be multiplied by the 3.
Next, we will multiply everything that is inside of the bracket by the a value, which is 3. The (x-2)^2 will be multiplied by the 3, and the -4 will be multiplied by the 3 as well.
Finally, we can finish this by adding like terms. We must the number we left outside of the bracket to the number that was multiplied by the a value. In this case, the number outside was -5, and the number we got after multiplying by 3 was -12.
The a is 3, the h is -2, and the k is -17.
Let's try another example. This example will have a negative a value, so we will need to factor out -1 from everything inside the brackets.
SOLVING QUADRATIC EQUATIONS
Solving Quadratic Equations by Factoring
In factored form, if the equation is equal to 0 (the y), that means that one or both of the factors is equal to 0. So solve for x, we will set each of the factors to 0, and isolate x. It should look like this:
Next, let's try an equation with a coefficient in front of x. Since the x is multiplied by the coefficient, the only new step is to divide both sides by the coefficient. Let's try an example:
Solving Quadratic Equations Using the Quadratic Formula
Also, look back at the formula. There is a plus and minus sign together. This means that the formula can be used twice: one with a plus in that spot and one with a minus in that spot. Since the formula can be used twice, each time will give a different root, for a total of 2 roots.
Let's try an example:
There is one more thing to bring to attention. Look back to the work for the solution. There is a square root in there. The number that is being square rooted is called a discriminant. In this quadratic equation, the discriminant was 52. The discriminant can help us know how many roots there will be.
- If the discriminant is positive, there will be 2 roots. This is because the number will be squared, and either added to or subtracted from the number before it.
- If the discriminant is negative, there will be no roots. This is because a negative number cannot be square rooted, as a square is a number times itself. To get a negative sign, the signs multiplied must be opposite, so you cannot square root a negative following this concept.
- If the discriminant is zero, there will be 1 root. This is because the square root of 0 is 0, so nothing will be added or subtracted to the number before it, and it will just be the number -b divided by 2(a).
Solving Quadratic Equations by Isolating X
Like the quadratic formula and the discriminant, there is also a way to find out how many roots we will have by looking at the equation. For vertex form, we must look at the a and k values:
- If a is positive, and k is positive, there will be 0 roots.
- If a is positive and k is negative, there will be 2 roots.
- If a is negative and k is positive, there will be 2 roots.
- If a is negative and k is negative, there will be 0 roots.
- If k is zero, there will be 1 root regardless of what the a is.
- Expanding binomials will help you get from factored and vertex form to standard form.
- Completing the square will get you from standard form to vertex form. This also requires factoring (perfect square trinomials).
- Factoring will also help you get from standard form to factored form.
- Factoring relates to graphing because it can help us get roots.
- We can use quadratic formula from standard form to get roots.
- Roots can help us get the vertex, and graph the parabola.
- Vertex form relates to graphing because it shows us the vertex and translations, and also the step pattern.
- Learning about discriminants will help us know how many roots there are.
- Algebra tiles can help with multiplying binomials and factoring.
b) Find the times when the tennis ball is at a height of 4.5 m above the ground. Round your answers to the nearest tenth of a second.
c) What is the maximum height of the tennis ball? At what time does it reach this height? Round your answers to the nearest tenth.
To answer these:
For a), we need to find the roots. For b), we need to find the roots when y=4.5. For c), we need to find the vertex. We can answer these questions using the quadratic formula.
Calculators are sold to students for 20 dollars each. 300 students are willing to buy them at that price. For every 5 dollar increase in price, there are 30 fewer students willing to buy the calculator. What selling price will produce the maximum revenue and what will the maximum revenue be?
To answer this:
We will first need to define our variables. Next, we will need to make our equation. We can use factored form for this. Then, we must find our roots and vertex.
a) Write an equation that can be used to find the length and width of the rectangle.
b) What are the dimensions of the rectangle?
To answer this:
First, define your variables. Next, create an equation (remember, area = l*w.) Then, solve for your variables. Here's an important tip: dimensions cannot be negative. This narrows your roots to only one. Also, drawing a diagram can help.
Consecutive Number Problems
To answer this:
First, define your variables. Next, create an equation. After this, you can solve the quadratic equation.
Right Triangle Problems (Pythagoras' Theorem)
To answer this:
Define variables. Use Pythagoras' Theorem. Base the lengths off of 1 variable. Solve for that variable. (Hint: a diagram can be helpful.)
Optimization Problem (Area)
To solve this:
First, define variables and create an equation. Next, you must find roots, axis of symmetry and the optimal value. Also, a diagram might help.
REFLECTION OF THE UNIT
Here is one of my assessments from this unit:
- For the matching part, I was able to identify the types of equations and which methods would be used from practice.
- Number 6 was multiplying binomials. I multiplied the 2 binomials, then added like terms.
- Number 7 was factoring. The first question was a complex trinomial. I found the factors by finding the factors of a and c, and making sure that when multiplied and added, they would add to b.
- Number 8 was factoring, and then solving for the roots. To do this, i factored each equation, then made it equal to 0. After this, I isolated the x's to find the 2 roots.
- In number 9, I once again used factoring to find the roots. Once I had the roots, I found the axis of symmetry and the optimal value, which gave me the vertex. Finally, I graphed it by plotting the vertex and the roots.
- In number 10, I created an equation for each shape. I made an equation to find the area of the square and an equation to find the area of the rectangle. I then subtracted the area of the rectangle from the area of the square, and added like terms. This gave me the area of the shaded region. I then factored it in the next question using common factoring.
- In number 11/12, I was able to find the answer that would give one root, which was 30. I got this because the trinomial in the question was a perfect square trinomial. When there is a perfect square trinomial, there is only 1 root, so I solved for the b of the equation.
All in all, this assessment was good. I got a great mark, and I learned a lot from this part. Factoring and multiplying binomials is an important concept to understand, and I'm proud that I did good on the test.
That's all that's left on this Smore. However, if there is something you need clarification on, or need more practice, you can visit: