Use the function f(x) to answer the questions
f(x) = 2x^2 - 3x - 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 a 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.

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Step-by-step answer

10.05.2023, solved by verified expert
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Answer:

Answers explained below.

Step-by-step explanation:

Part A. 

For x-intercepts, 

2x² - 3x - 5 = 0

2x² + 2x - 5x - 5 = 0

2x(x + 1) - 5(x + 1) = 0

(2x - 5)(x + 1) = 0

2x - 5 = 0 and x + 1 = 0

x = 2.5 and x = -1

Part B. 

As the coefficient of the x² term is positive, the parabola open up, and the vertex is a minimum. The coordinates of the parabola are given by -[b/2a, f(-b/2a)] where a and b represent the coefficients of a parabola in the standard form ax² + bx +c = 0. 

In this case a = 2, b = -3, c = -5. 

So -b/2a = -(-3)/(2 × 2) 

= 3/4 and 

f(-b/2a) = 2(3/4)² - 3(3/4) - 5

= 2(9/16) - 9/4 - 5

= -6.125

The coordinates of the vertex are (0.75, -6.125)

Part C. 

To graph this function, 

Firstly, establish where the vertex will be from the coordinates found. 

Next, plot the position of the x intercepts. You know it opens up, so you would draw it so the vertex is a minimum. 

You could also get the y intercept by letting x = 0, which gives -5 for the y intercept. This should give you a rough idea of what the graph looks like.

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