Question

use your graphing calculator to sketch the graph of the function, then determine the coordinates of the x-intercepts for the function, if they exist.

$y = -2x^2 + 5x - 2$

graph the function using a graphing calculator.

a.
−10,10 by −20,20, xscl=1, yscl=2
b.
−10,10 by −20,20, xscl=1, yscl=2
c.
−10,10 by −20,20, xscl=1, yscl=2
d.
−10,10 by −20,20, xscl=1, yscl=2

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the x-intercept(s) is/are
(type an ordered pair, using integers or decimals. use a comma to separate answers as needed.)
b. there is no x-intercept.

Explanation:

Step 1: Recall the formula for x - intercepts

To find the x - intercepts of a function \(y = f(x)\), we set \(y=0\) and solve for \(x\). For the quadratic function \(y=-2x^{2}+5x - 2\), we set \(y = 0\), so we have the equation \(-2x^{2}+5x - 2=0\). Multiply both sides by - 1 to get \(2x^{2}-5x + 2 = 0\).

Step 2: Factor the quadratic equation

We factor the quadratic \(2x^{2}-5x + 2\). We need two numbers that multiply to \(2\times2 = 4\) and add up to - 5. The numbers are - 1 and - 4. So we can rewrite the middle term: \(2x^{2}-x-4x + 2=0\). Group the terms: \((2x^{2}-x)-(4x - 2)=0\). Factor out the common factors from each group: \(x(2x - 1)-2(2x - 1)=0\). Then we have \((2x - 1)(x - 2)=0\).

Step 3: Solve for x

Set each factor equal to zero:

• For \(2x-1 = 0\), add 1 to both sides: \(2x=1\), then divide by 2: \(x=\frac{1}{2}=0.5\).
• For \(x - 2=0\), add 2 to both sides: \(x = 2\).

Since the x - intercepts are the points where \(y = 0\), the x - intercepts are \((0.5,0)\) and \((2,0)\).

Answer:

A. The x - intercept(s) is/are \((0.5,0),(2,0)\)