Question

secondary math ii // module 2
structures of expressions - 2.9
set
topic: factored form of a quadratic function
given the factored form of a quadratic function, identify the vertex, intercepts, and vertical stretch of the parabola.

1. $y = 4(x - 2)(x + 6)$

a. vertex:____
b. x - inter(s)____
c. y - inter____
d. stretch____

1. $y=-3(x + 2)(x - 6)$

a. vertex:____
b. x - inter(s)____
c. y - inter:____
d. stretch____

1. $y=(x + 5)(x + 7)$

a. vertex:$(-6,-1)$
b. x - inter(s)$(-5,0),(-7,0)$
c. y - inter$(0,35)$
d. stretch$a = 1$

1. $y=\frac{1}{2}(x - 7)(x - 7)$

a. vertex:____
b. x - inter(s)____
c. y - inter____
d. stretch____

1. $y=-\frac{1}{2}(x - 8)(x + 4)$

a. vertex:____
b. x - inter(s)____
c. y - inter:____
d. stretch____

1. $y=\frac{3}{5}(x - 25)(x - 9)$

a. vertex:____
b. x - inter(s)____
c. y - inter____
d. stretch____

1. $y=\frac{3}{4}(x - 3)(x + 3)$

a. vertex:____
b. x - inter(s)____
c. y - inter____
d. stretch____

1. $y=-(x - 5)(x + 5)$

a. vertex:____
b. x - inter(s)____
c. y - inter:____
d. stretch____

1. $y=\frac{2}{3}(x + 10)(x + 10)$

a. vertex:____
b. x - inter(s)____
c. y - inter____
d. stretch____

Explanation:

Step1: Find x - intercepts

Set $y = 0$. For $y=a(x - p)(x - q)$, the x - intercepts are $x=p$ and $x = q$.

Step2: Find y - intercept

Set $x = 0$. Substitute $x = 0$ into $y=a(x - p)(x - q)$ to get $y=a(0 - p)(0 - q)=apq$.

Step3: Find vertex x - coordinate

The x - coordinate of the vertex of $y=a(x - p)(x - q)$ is $x=\frac{p + q}{2}$. Then substitute this x - value into the function to get the y - coordinate of the vertex.

Step4: Identify stretch factor

The stretch factor is the coefficient $a$ of the factored - form quadratic function $y=a(x - p)(x - q)$.

For $y = 4(x - 2)(x + 6)$:

- **a. Vertex**:
- The x - coordinate of the vertex $x=\frac{2+( - 6)}{2}=\frac{2 - 6}{2}=-2$.
- Substitute $x=-2$ into $y = 4(x - 2)(x + 6)$: $y=4(-2 - 2)(-2 + 6)=4\times(-4)\times4=-64$. So the vertex is $(-2,-64)$.
- **b. x - intercepts**:
- Set $y = 0$. Then $4(x - 2)(x + 6)=0$, so $x = 2$ and $x=-6$.
- **c. y - intercept**:
- Set $x = 0$. Then $y=4(0 - 2)(0 + 6)=4\times(-2)\times6=-48$.
- **d. Stretch**: The stretch factor $a = 4$.

For $y=-3(x + 2)(x - 6)$:

- **a. Vertex**:
- The x - coordinate of the vertex $x=\frac{-2 + 6}{2}=2$.
- Substitute $x = 2$ into $y=-3(x + 2)(x - 6)$: $y=-3(2 + 2)(2 - 6)=-3\times4\times(-4)=48$. So the vertex is $(2,48)$.
- **b. x - intercepts**:
- Set $y = 0$. Then $-3(x + 2)(x - 6)=0$, so $x=-2$ and $x = 6$.
- **c. y - intercept**:
- Set $x = 0$. Then $y=-3(0 + 2)(0 - 6)=-3\times2\times(-6)=36$.
- **d. Stretch**: The stretch factor $a=-3$.

For $y=\frac{1}{2}(x - 7)(x - 7)=\frac{1}{2}(x - 7)^2$:

- **a. Vertex**:
- The x - coordinate of the vertex is $x = 7$.
- Substitute $x = 7$ into $y=\frac{1}{2}(x - 7)^2$, $y = 0$. So the vertex is $(7,0)$.
- **b. x - intercepts**:
- Set $y = 0$. Then $\frac{1}{2}(x - 7)^2=0$, so $x = 7$.
- **c. y - intercept**:
- Set $x = 0$. Then $y=\frac{1}{2}(0 - 7)^2=\frac{1}{2}\times49=\frac{49}{2}$.
- **d. Stretch**: The stretch factor $a=\frac{1}{2}$.

For $y=-\frac{1}{2}(x - 8)(x + 4)$:

- **a. Vertex**:
- The x - coordinate of the vertex $x=\frac{8+( - 4)}{2}=2$.
- Substitute $x = 2$ into $y=-\frac{1}{2}(x - 8)(x + 4)$: $y=-\frac{1}{2}(2 - 8)(2 + 4)=-\frac{1}{2}\times(-6)\times6 = 18$. So the vertex is $(2,18)$.
- **b. x - intercepts**:
- Set $y = 0$. Then $-\frac{1}{2}(x - 8)(x + 4)=0$, so $x = 8$ and $x=-4$.
- **c. y - intercept**:
- Set $x = 0$. Then $y=-\frac{1}{2}(0 - 8)(0 + 4)=-\frac{1}{2}\times(-8)\times4 = 16$.
- **d. Stretch**: The stretch factor $a=-\frac{1}{2}$.

For $y=\frac{3}{5}(x - 25)(x - 9)$:

- **a. Vertex**:
- The x - coordinate of the vertex $x=\frac{25 + 9}{2}=17$.
- Substitute $x = 17$ into $y=\frac{3}{5}(x - 25)(x - 9)$: $y=\frac{3}{5}(17 - 25)(17 - 9)=\frac{3}{5}\times(-8)\times8=-\frac{192}{5}$. So the vertex is $(17,-\frac{192}{5})$.
- **b. x - intercepts**:
- Set $y = 0$. Then $\frac{3}{5}(x - 25)(x - 9)=0$, so $x = 25$ and $x = 9$.
- **c. y - intercept**:
- Set $x = 0$. Then $y=\frac{3}{5}(0 - 25)(0 - 9)=\frac{3}{5}\times(-25)\times(-9)=135$.
- **d. Stretch**: The stretch factor $a=\frac{3}{5}$.

For $y=\frac{3}{4}(x - 3)(x + 3)$:

- **a. Vertex**:
- The x - coordinate of the vertex $x=\frac{3+( - 3)}{2}=0$.
- Substitute $x = 0$ into $y=\frac{3}{4}(x - 3)(x + 3)$: $y=\frac{3}{4}(0 - 3)(0 + 3)=\frac{3}{4}\times(-3)\times3=-\frac{27}{4}$. So the vertex is $(0,-\frac{27}{4})$.
- **b. x - intercepts**:
- Set $y = 0$. Then $\frac{3}{4}(x - 3)(x + 3)=0$, so $x = 3$ and $x=-3$.
- **c. y - intercept**:
- Set $x = 0$. Then $y=\frac{3}{4}(0 - 3)(0 + 3)=-\frac{27}{4}$.
- **d. Stretch**: The stretch factor $a=\frac{3}{4}$.

For $y=-(x - 5)(x + 5)$:

- **a. Vertex**:
- The x - coordinate of the vertex $x=\frac{5+( - 5)}{2}=0$.
- Substitute $x = 0$ into $y=-(x - 5)(x + 5)$: $y=-(0 - 5)(0 + 5)=25$. So the vertex is $(0,25)$.
- **b. x - intercepts**:
- Set $y = 0$. Then $-(x - 5)(x + 5)=0$, so $x = 5$ and $x=-5$.
- **c. y - intercept**:
- Set $x = 0$. Then $y=-(0 - 5)(0 + 5)=25$.
- **d. Stretch**: The stretch factor $a=-1$.

For $y=\frac{2}{3}(x + 10)(x + 10)=\frac{2}{3}(x + 10)^2$:

- **a. Vertex**:
- The x - coordinate of the vertex is $x=-10$.
- Substitute $x=-10$ into $y=\frac{2}{3}(x + 10)^2$, $y = 0$. So the vertex is $(-10,0)$.
- **b. x - intercepts**:
- Set $y = 0$. Then $\frac{2}{3}(x + 10)^2=0$, so $x=-10$.
- **c. y - intercept**:
- Set $x = 0$. Then $y=\frac{2}{3}(0 + 10)^2=\frac{2}{3}\times100=\frac{200}{3}$.
- **d. Stretch**: The stretch factor $a=\frac{2}{3}$.

Answer:

11. $y = 4(x - 2)(x + 6)$

a. $(-2,-64)$
b. $x = 2,x=-6$
c. $-48$
d. $4$

12. $y=-3(x + 2)(x - 6)$

a. $(2,48)$
b. $x=-2,x = 6$
c. $36$
d. $-3$

14. $y=\frac{1}{2}(x - 7)(x - 7)$

a. $(7,0)$
b. $x = 7$
c. $\frac{49}{2}$
d. $\frac{1}{2}$

15. $y=-\frac{1}{2}(x - 8)(x + 4)$

a. $(2,18)$
b. $x = 8,x=-4$
c. $16$
d. $-\frac{1}{2}$

16. $y=\frac{3}{5}(x - 25)(x - 9)$

a. $(17,-\frac{192}{5})$
b. $x = 25,x = 9$
c. $135$
d. $\frac{3}{5}$

17. $y=\frac{3}{4}(x - 3)(x + 3)$

a. $(0,-\frac{27}{4})$
b. $x = 3,x=-3$
c. $-\frac{27}{4}$
d. $\frac{3}{4}$

18. $y=-(x - 5)(x + 5)$

a. $(0,25)$
b. $x = 5,x=-5$
c. $25$
d. $-1$

19. $y=\frac{2}{3}(x + 10)(x + 10)$

a. $(-10,0)$
b. $x=-10$
c. $\frac{200}{3}$
d. $\frac{2}{3}$