# Standard Form

## Learning Goals

I am able to:

1. Determine the key features of a parabola from a quadratic equation given in standard form
2. Graph a quadratic relation given in standard form
3. Determine the number of zeros a quadratic relation has by finding the discriminant
4. Use the quadratic formula to solve the quadratic relation given in standard form
5. Convert a standard form equation to vertex form by completing the squares
6. Solve word problems for quadratic relations in standard form

## Summary of Standard Form

There are three forms of a quadratic equation: Vertex form, factored form and standard form. In this unit, we will be looking at the quadratic equation y = ax²+bx+c, which is in standard form.

This unit explores how you can determine the key features of a parabola from a quadratic equation given in standard form, how to graph a quadratic relation given in standard form, how to determine the number of zeros by finding the discriminant, use the quadratic formula to solve standard form equations, use the completing the squares method to convert a standard form equation to vertex form, and solve word problems for quadratic relations in standard form.

The key features that can be found from a quadratic equation in standard form are vertex, shape, direction of opening, y-intercept, x-intercepts, axis of symmetry and minimum/maximum value.

I can then use these key features to graph the parabola of the standard form equation.

A quadratic relation can have 0, 1, or 2 zeros. This can be determined by solving the discriminant.

## Determining Key Features of a Parabola

The key features of a parabola are:

• Vertex - the point at which the parabola changes direction

• Shape - the value of a tells us if the parabola is stretched or compressed (if a>0, it is stretched; if a<0, it is compressed)
• Direction of Opening - the value of a tells us if the parabola opens up or down (if a is negative, it opens down; if a is positive, it opens up)
• y-intercept - the point at which the parabola passes through the y-axis (the value of c is the y-intercept)
• Zeros (x-intercepts) - the point(s) at which the parabola passes through the x-axis
• Axis of Symmetry - a vertical line passing through the vertex, making both sides look exactly the same
• Minimum/Maximum Value - the y-coordinate of the vertex (if a<0, it is maximum value; if a>0, it is minimum value)

e.g. y=3x²+6x-4

a) Find the x-intercepts.

b) Find the vertex.

c) Write the equation in vertex form.

## Graphing Standard Form Equations

In order to graph an equation that is given in standard form, you need to find three points. These include the x-intercepts, the vertex and the y-intercept.

Graph the following equation: y=x²-6x+5.

How to Graph Linear Equations: Standard Form (Cover-up Method)

## Determining the Number of Zeros

A discriminant is a number that tells you how many solutions there are to an equation.

In order to determine the number of solutions/zeros, you have to use the formula b²-4ac, which originates from the quadratic formula. All you have to do is substitute the values of the variables given in the standard form equation.

As you can tell, if the value of the discriminant is less than 0 (negative), there are no solutions to the equation. If the value of the discriminant is equal to 0, there is one solution to the equation. Finally, if the value of the discriminant is greater than 0 (positive), there are two solutions for the equation.

We can use the quadratic formula to solve for the x-intercepts. The formula is:

e.g. Find the x-intercepts of y=4x²+2x-3.

The first step to solving this problem is to substitute the values of a, b, and c into the quadratic formula.

a=4, b=2, c=-3

Then, we continue to solve for x.
The ± sign stands for plus or minus.

This means that after we solve for x, we have to find the exact x values by adding for one, and then subtracting for the other x value.

How to use the quadratic formula | Polynomial and rational functions | Algebra II | Khan Academy
The vertex can be found from the x-intercepts as well. The x-value of the vertex (axis of symmetry) is determined by adding both x-intercepts and then dividing by 2. The y-value of the vertex (minimum/maximum) can be found by substituting the x-value into the original equation.

## How to Convert Standard Form Equation to Vertex Form

When you convert a standard form equation to vertex form, you have to use a method called completing the squares.

e.g. 3x²+6x-7=0

Step 1: Put brackets around the values of ax² and bx and keep the constant value outside.

(3x²+6x)-7=0

Step 2: Take the coefficient of the value of a outside of the bracket and divide the value

of b by a. 3(x²+2x)-7=0

Step 3: Divide the new value of b by 2 and square that number. Add that number once and subtract it once. 3(x²+2x+1-1)-7=0

Step 4: Take the negative value of the number from step 3 outside of the bracket by multiplying it with the coefficient in front of the bracket. 3(x²+2x+1)-3-7=0

Step 5: Add the numbers after the brackets. 3(x²+2x+1)-10=0

Step 6: Factor out the values in the brackets. 3(x+1)²-10=0

Solving quadratic equations by completing the square | Algebra II | Khan Academy

## Word Problems

1. A soccer ball is thrown from the top of a building and falls to the ground. Its path is represented by the relation: h=-5t²+20t+2, where h represents the height above the ground in metres and t represents the times in seconds.

a) How tall is the building?

b) What is the maximum height of the soccer ball?

c) At what time does the soccer ball reach its maximum height?

2. The sum of the squares of two consecutive integers is 365. What are the two integers?

Let represent the first consecutive integer and let (x+2)² represent the second consecutive integer.

Check:

13+1=14

-14+1=-13

Therefore, the two consecutive integers are 13 and 14.

"Math is everywhere", is what they say. Every part of your life, even the smallest things, involve the use of math. Therefore, it is very important to understand and comprehend the key concepts. This year, one of the main units we have covered is Quadratic Relations. This unit explores the three forms of a quadratic relation, which are: vertex form (y=a(x-h)²+k), factored form (y=a(x-r)(x-s)), and standard form (y=ax²+bx+c).

For vertex form, I learned how to describe the characteristics of a parabola (e.g. x-intercepts, vertex, y-intercept), determine the finite differences of a quadratic relation (e.g. linear, non-linear, neither), create a parabola, describe the transformations of a parabola and solve quadratic word problems. Even though this was probably the easiest part of the Quadratics Unit, I had some ups and downs with it because I found it difficult to solve word problems related to it.

For factored form, I learned how to expand and simplify in order to solve factored form expressions, use various factoring methods to solve factored form equations (e.g. binomial common factoring, factor by grouping, complex trinomial factoring, difference of squares and perfect square trinomial), determine key features of a parabola from a factored form equation, create a parabola from an equation in factored form and solve factoring word problems. This unit was the easiest for me because I enjoyed solving equations and I started to get a better understanding of the word problems. I comprehended it very well and grasped the concepts quickly.

For standard form, I learned how to determine the key features of a parabola from a quadratic equation given in standard form, graph the quadratic relation, determine the number of zeros by finding the discriminant (b²-4ac), use the quadratic formula to solve the standard form equation and solve standard form word problems. In this unit, I had little difficulty as I was unsure of which method I should use to solve the word problems (quadratic formula or completing the squares). Other than that, I had a pretty good understanding of the concepts that we learned.

I was able to connect my learning of standard form with vertex form by converting it using the completing the squares method. In order to do this, I had to first put brackets around the values of ax² and bx and keep the constant value outside. Then, I had to take the coefficient of the value of a outside of the brackets and divide the value of b by a. Next, I needed to divide the new value of b by 2 and square that number, and then add that number once and subtract it once. After that, I had to take the negative value of the number from the last step outside of the bracket by multiplying it with the coefficient in front of the bracket. Then, I needed to add the numbers after the brackets. Finally, the last step was to factor out the values in the brackets.

In the beginning, I did have some difficulty adjusting to the new unit and solving word problems related to it, so I wasn't getting the marks I wanted. However, I continued to practice the things that weren't clear to me and now I am able to completely understand those concepts thoroughly. I realized that "the only way to learn mathematics is to do mathematics." -Paul Halmos.