Question

mr richardson, math 2 (1111 b), convert factored form to standard form, 12/9/25
name jonathan francis period 2

1. use quadratic equation in factored form: f(x) = (-2x + 2)(x - 4)

a) convert the given equation to standard form?
f(x) = -2x² - 6x - 8
b) what are the coordinates of the y - intercept? (0, -8)
c) what are the coordinates of each of the x - intercepts? set each given factor to zero. put answers here: 1) (1, 0) 2) (4, 0)
d) what is the x - coordinate of the vertex? -1.5
show work here: x = -b/(2a), x = -(-6)/(2*(-2)) = 6/(-4) = -1.5
e) what is the y - coordinate of the vertex?
show work here: f(-1.5) = -2(-1.5)² - 6(-1.5) - 8, f(-1.5) = -2(2.25) - 6(-1.5) - 8 = -4.5 + 9 - 8

Explanation:

Step1: Expand the factored form

$$f(x) = (-2x + 2)(x - 4) = -2x(x - 4) + 2(x - 4) = -2x^2 + 8x + 2x - 8 = -2x^2 + 10x - 8$$

Step2: Find y-intercept (x=0)

$$f(0) = -2(0)^2 + 10(0) - 8 = -8$$

Step3: Find x-intercepts (f(x)=0)

Set factors to zero: $-2x + 2 = 0 \Rightarrow x=1$; $x - 4 = 0 \Rightarrow x=4$

Step4: Find vertex x-coordinate

Use $x = -\frac{b}{2a}$ where $a=-2, b=10$: $x = -\frac{10}{2(-2)} = 2.5$

Step5: Find vertex y-coordinate

Substitute $x=2.5$ into $f(x)$: $f(2.5) = -2(2.5)^2 + 10(2.5) - 8 = -12.5 + 25 - 8 = 4.5$

Answer:

a) $f(x) = -2x^2 + 10x - 8$
b) $(0, -8)$
c) $(1, 0), (4, 0)$
d) $2.5$
e) $4.5$