Question

Use the function f(x) to answer the questions:

f(x) = 4x2 + 8x − 5

Part A: What are the x-intercepts of the graph of f(x)? Show your work. (2 points)

Part B: Is the vertex of the graph of f(x) going to be a maximum or a minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points)

Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)

Answer 1

Explanation:

let's start with B :

the best way to find the vertex is by transforming the function into the vertex form :

y = a(x - h)² + k

with (h, k) being the vertex.

when doing all the multiplications, we get

y = a(x²-2hx+h²) + k = ax² - 2ahx + ah² + k

let's compare with the irreducible function

ax² = 4x²

a = 4

-2ahx = 8x

-2×4hx = 8x

-8hx = 8x

-8h = 8

-h = 1

h = -1

ah² + k = -5

4×(-1)² + k = -5

4 + k = -5

k = -9

so, the vertex form is

y = 4(x + 1)² - 9

the vertex is (-1, -9)

if the factor of the (x-h)² term is greater than 0, then the parabola opens upward, and if it is less than zero, the parabola opens downward.

out factor is +4, so it opens upward, and the vertex is a minimum.

A

now we know

y = 4(x + 1)² - 9

for the x-intercepts we need to find the x-values that lead to 0 y values.

so,

0 = 4(x + 1)² - 9

9 = 4(x + 1)²

3 = ± 2(x + 1) = ± (2x + 2)

for +(2x + 2) is then

3 = 2x + 2

2x = 1

x = 1/2

for -(2x + 2) is then

3 = -2x - 2

5 = -2x

x = -5/2

so, the x-intercepts are

(-5/2, 0) and (1/2, 0).

C

I know it is a parabola.

i know 3 points : the vertex and the 2 x-interceot points. i could then also calculate the points for x = 0, 1, -1, 2, -2.

and then draw the graph with a curve ruler connecting the points.