a Set each equation equal to zero, then replace the zero with a y to get the related function.
B
b Use the graphing function on the calculator, change the window to see the full parabola.
C
c The table feature on the calculator is found by pressing the 2nd key and then the GRAPH key. Scroll up and down the table until you find the place where the y-value is zero.
D
d Notice how the y-values compare to each other around the zeros of the same function.
A
a
Equation
Related Function
x^2-x=12
y = x^2-x-12
x^2+8x=9
y = x^2+8x-9
x^2=14x-24
y = x^2-14x+24
x^2+16x=-28
y = x^2+16x+28
B
b
C
c
Equation
Related Function
Zeros
y-Values
x^2-x=12
y = x^2-x-12
x=-3 x=4
y=8 and y=-6 y=8 and y=-6
x^2+8x=9
y = x^2+8x-9
x=-9 x=1
y=-9andy=11 y=-9andy=11
x^2=14x-24
y = x^2-14x+24
x=12 x=2
y=-9andy=11 y=-9andy=11
x^2+16x=-28
y = x^2+16x+28
x=-14 x=-2
y=-11andy=13 y=-11andy=13
D
d The function values are the same around each zero and there is always one positive and one negative function value.
Practice makes perfect
a To convert a quadratic equation to a function, we need to rearrange the equation so that it is equal to zero. Then we can replace the zero with y to get the function. Let's look at one equation at a time.
x^2-x = 12
⇕
x^2-x-12 = 0
We needed to subtract 12 from both sides of our first equation. We can replace the 0 with a y to get our related function, y = x^2 -x - 12. Let's repeat this process for the second function.
x^2+8x = 9
⇕
x^2+8x-9 = 0
We needed to subtract 9 from both sides of our second equation. We can replace the 0 with a y to get our related function, y = x^2 +8x - 9. Let's repeat this process for the third function.
x^2=14x - 24
⇕
x^2-14x= -24
⇕
x^2-14x+24=0
We needed to subtract 14x from both sides and then add 24 to each side of our third equation. We can replace the 0 with a y to get our related function, y = x^2 -14x + 24. Let's repeat this process for the fourth function.
x^2+16x = -28
⇕
x^2+16x+28 = 0
We needed to add 28 to both sides of our fourth equation. We can replace the 0 with a y to get our related function, y = x^2 +16x +28. Let's put these answers in our table.
Equation
Related Function
Zeros
y-Values
x^2-x=12
y = x^2-x-12
x^2+8x=9
y = x^2+8x-9
x^2=14x-24
y = x^2-14x+24
x^2+16x=-28
y = x^2+16x+28
b Now that the equations are in function form, we can use our calculators to graph them. First, let's plot the function. We can press Y= to enter our first function. Then, press GRAPH to see the graph. You may have to adjust the window to see the full graph.
Let's repeat this for our second function.
Let's do it again for our third function.
Finally, let's plot our fourth function on the calculator.
c This exercise asks us to use the table function to find the zeros. Since we also need to look at the values above and below each zero, this will give us more than just the zeros. To get a table, use the function and the Y= as before, but this time press 2nd before pressing the button marked GRAPH for the table.
Our first zero is x=-3 and the y-values just before and after it are 8 and -6. Let's scroll down the list to find the second zero.
From this part of the table, we see the second zero is at x=4. The values before and after the zero are y=8 and y=-6. Let's repeat this process for our second function.
Our first zero is x=-9 and the y-values just before and after it are y=11 and y=-9. Let's scroll down the list to find the second zero.
From this part of the table, we see the second zero is at x=1. The values before and after the zero are y=-9 and y=11. Let's repeat this process again for the third function.
Our first zero is x=2 and the y-values just before and after it are y=11 and y=-9. Let's scroll down the list to find the second zero.
From this part of the table, we see the second zero is at x=12. The values before and after the zero are y=-9 and y=11. Let's repeat this process for our last function.
Our first zero is x=-14 and the y-values just before and after it are y=13 and y=-11. Let's scroll down the list to find the second zero.
From this part of the table, we see the second zero is at x=-2. The values before and after the zero are y=-11 and y=13. Now, let's add all this information to the table.
Function
Related Function
Zeros
y-Values
x^2-x=12
y = x^2-x-12
x=-3 x=4
y=8 and y=-6 y=8 and y=-6
x^2+8x=9
y = x^2+8x-9
x=-9 x=1
y=-9andy=11 y=-9andy=11
x^2=14x-24
y = x^2-14x+24
x=12 x=2
y=-9andy=11 y=-9andy=11
x^2+16x=-28
y = x^2+16x+28
x=-14 x=-2
y=-11andy=13 y=-11andy=13
d In Part C, we listed the function values before and after the zero. We can see that the function values are the same around each zero. It is also noteworthy that there is always one positive and one negative function value around each zero.
