PSAT Math Grid-Ins Question 16: Answer and Explanation

Question: 16

In the above graph, parabola f(x) is represented by the equation f(x) = x² - 4 and line g(x) is represented by the equation g(x)= -x -4. Line g(x) intersects parabola f(x) at point (0, -4).

A new line, h(x), is added to the graph. The line h(x) is perpendicular to line g(x), intersecting with line g(x) at the point (-12, 0). What is the x-coordinate of the point where line h(x) will intersect parabola f(x) in Quadrant 1?

Correct Answer: 8

Explanation:

8

The question asks for an x-coordinate. The question provides a lot of information, so take it in Bite-Sized Pieces. Line h(x) is perpendicular to line g(x) and the two intersect at the point (-12, 0). Perpendicular lines have negative reciprocal slopes. Because the equation for line g(x) is in slope-intercept form, the slope of line g(x) is the coefficient of x (the m-value) in the given equation, or . The negative reciprocal of is 3, so the slope of h(x) is 3. The equation for h(x) then becomes h(x) = 3x + b. Plug in (-12, 0) to solve for b. The equation becomes 0 = 3(-12) + b, or 36 = b. Therefore, h(x) = 3x + 36. To find the point at which h(x) intersects f(x), set the function equations equal to one another: x² - 4 = 3x + 36. To solve for x, turn this into a quadratic equation by moving all terms to the left side and combining like terms: x² - 3x - 40 = 0. This equation factors to (x + 5)(x - 8) = 0. Therefore, the two solutions are x = -5 and x = 8. The question asks for a solution that is in Quadrant I—the upper right quadrant. In this quadrant, both x and y are positive, so x = 8 is the solution in this quadrant. The correct answer is 8.