Practise makes perfect!
1. Parabolas: We have seen lines on a graph, when a curve is formed on a graph it's called a parabola.
2. Vertex: Vertex is the highest point on a graph.
3. Zeros: Zeros are basically the x intercepts. The x intercept is the point where the line touches the x axis on a point. The x intercepts/ zeros are also the x value of the vertex.
4. Optimal Value: Optimal value is the y value of the vertex.
5. Maximum Value: When any equation/ word problems tells us to find the maximum value of something, they are referring to the vertex.
The table and the graph on the left show a linear relationship. The relationship is linear because the difference between the y values is 4 and its constant.
The table of values on the left shows that the y value is not constant as it does not increase by the same number, therefore we know that the relationship is not linear. However, what one can see is that the first difference numbers increase by the same number which is 2. When the difference between the first differences are constant, we call that second differences. If the second difference is the same, we know the relationship is quadratic not linear.
When the second differences are the same, the relationship is quadratic. Just like a linear relationship forms a straight line on the graph, a quadratic relationship forms a curve on the graph. The curve is called a parabola.
Three types of equations
- The first type of equation is written in vertex form. y=a(x-h)²+k is the basic equation written in vertex form.
- Next equation is written in factored form. y=a(x-r)(x-s) is the basic equation.
- The last type of equation is called standard form. y=ax²+bx+c is the equation we can use to go from vertex to factored form.
Parabolas/ Vertex form
The a value in front of the equation tells us whether the parabola opens up or down. If the a value is a positive whole number of a fraction, the parabola will open up. When the a value is a whole number, its called a vertical stretch. If the a value is a fraction, its parabola's transition is described as a vertical compression.
The picture below shows the shape of the parabola when it opens up. An easy way to remember this is the shape of a U.
An easy way to remember when a parabola opens down is when the parabola forms a downward U shape.
Graphing from Factored Form
y=0.5(x+3)(x+5) is an example of an equation written in factored form.
To graph a parabola, we need to find three peices of information.
1) State the zeros. ( This can be found by setting each factor equal to zero)
2) Find the axis of symmetry. ( This is where we add the two values of zero and divide the sum by 2 since we are finding the midpoint)
3) State the optimal value.( This is where we sub in the value of the axis of symmetry into the equation).
x= -3 x=-5
Therefore, the two x intercepts are (-3,0) and (-5.0)
2. Axis of symmetry= (-3)+(-5)/2
Now we found the x value of out the x value of the vertex which is (-4,0)
y=0.5( -4 +3)( -4 +5)
Now that we found the y value of the vertex (-0.5), we can graph it.
When a parabola has 2 x intercepts, it touches two points on the x axis. The points can negative or positive. When a parabola only has one x intercept, its the vertex that's the x intercept. A parabola can also have no x intercepts, this is when the vertex is above or below the x axis.
(x+2) ( x+5) is an example.
1. We can draw arrows to help us. Draw an arrow from the first x to the second x in the bracket. Next draw an arrow from the first x to 5 in the second bracket. When the multiply the two x's together, the answer is x² since there are two x's. When x and 5 are being multiplied we get 5x as the answer.
2. We also have to look at the second number in the first bracket which is 2. Multiply 2 by x and by 5. After multiplying these numbers, we get 2x and 10.
3. Join the answers together.
(x+2) ( x+5)
= x²+ 5x+2x+10
4. From this answer, check to see like terms and solve.
(x+2) ( x+5)
= x²+ 5x+2x+10
5. x²+7x+10 is the final answer as there are no like terms.
Remember we can have any letter in front of the number, and in this example, two n's are multiplied together so it creates n².
Complete the Square
The reason why there are two signs ( negative and positive) after the b value is because we can have two x intercepts, meaning two solutions.
Example#1 : 0= 5x²+3x+2
= -31 Therefore the -31 is the number inside the square root, and since we cannot square root a negative number, there is no solution to this equation meaning no x intercept.
This formula comes in handy when you have to graph a parabola. By checking the discriminant, one can already know how many x intercepts there will be.
Example #2 : y= 2x²+5x+3
(5)²- 4 (2)(3)
Therefore 9 is the number inside the square root and we know that this equation will have two solutions because the 9 can be positive or negative.
Area of figure 1: ( 3x+4) ( 3x+4) Step 1 : Expand to multiply the expressions
9x²+12x+12x+16 Step2: After solving, collect like terms
9x²+24x+16 This is the area of figure 1.
Area of figure 2: (x-5) (x-5) Step 1 : Expand to multiply the expressions
x²-5x-5x+25 Step 2 : Collect like terms from this point.
x²-10x+25 This is the area of the figure 2
To find the area of the shaded region, all we do is subtract figure one's area with figure 2.
(9x²+24x+16) - ( x²-10x+25 )
Remember to only subtract the like terms together
Write a conclusion statement
Therefore, the area of the shaded region is 8x²+34x-9
Revenue Word Problems
There are also multiply ways to graph a parabola, we can use vertex form, standard, or factored form equation. One can go from standard to vertex and to the factored form easily. The easiest way to graph can be using the factored form equations because we can find the x intercepts but setting each expressions equal to 0, then find axis of symmetry and the optimal value. One can also go from the vertex to factored form and then graph.
There are also multiple ways to solving an equation. Any types of factoring, expanding or the quadratic formula can be used.
We can also find the vertex by solving the equation by completing a square.
Even in word problems, one might have to factor and then use the quadratic formula.
Everything is linked and fits into pieces! Always look for the multiple ways to solve any problem