Question

An expression is shown below:

f(x) = 4x2 + 8x − 5

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

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

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.

Answer 1

Given function:

• f(x) = 4x² + 8x - 5

Part A

The x-intercepts are the points when f(x) = 0.

Set up equation and solve:

• 4x² + 8 x - 5 = 0
• x = ( - 8 ± √(8² + 4*4*5))/8
• x = (- 8 ± √144)/8
• x = ( - 8 ± 12)/8
• x = - 20/8 = - 2.5 and x = 4/8 = 0.5

The x-intercepts are (-2.5, 0) and (0.5, 0)

Part B

The vertex form of the quadratic equation:

• y = a(x - h)² + k, where (h, k) is the vertex

The vertex is minimum if the leading coefficient is positive and maximum if the leading coefficient is negative. The given equation has a = 4, hence its vertex is minimum.

Convert the given equation into vertex form:

• y = 4x² + 8x - 5
• y = 4x² + 8x + 4 - 4 - 5
• y = 4(x² + 2x + 1) - 9
• y = 4(x + 1)² - 9

As per equation the vertex is (- 1, - 9)

Part C

From the previous parts we found the two x-intercepts and vertex, also found that the vertex is the minimum, so the graph opens up.

Plot the points and connect to obtain the graph.

See below.