Chapter 8.5, Problem 33E

To determine

Tomatch :The function with its graph.

Answer to Problem 33E

The graph that is matched with the given function isC

Explanation of Solution

Given function: $y=\left(x-5\right)\left(x+3\right)$

Given graphs:

  

Calculation:

We know that equation $y=a{\left(x-h\right)}^2+k$ represents an upward parabola with vertex at point $\left(h,k\right)$ if $a>0$ and represents a downward parabola with vertex at point $\left(h,k\right)$ if $a<0$ .

Given quadratic function can be written as

  $\begin{array}{l}y=\left(x-5\right)\left(x+3\right)\\ y=x^2-2x-15\\ y=\left(x^2-2x+1\right)-16\\ y={\left(x-1\right)}^2-16\end{array}$

Comparing it with $y=a{\left(x-h\right)}^2+k$ , we get

  $a=1,h=1,k=-16$

Therefore, given quadratic function represents an upward parabola with vertex at point (1,-16)

The x-intercepts of the curve are points where $y=0$

  $\begin{array}{l}0=\left(x-5\right)\left(x+3\right)\\ x=-3,5\end{array}$

That is x-intercepts are the points (-3,0) and (5,0).

The y-intercept is the point where x=0

  $y=\left(0-5\right)\left(0+3\right)=-15$

That is y-intercept of the curve is (0,-15).

Therefore, graph of quadratic function is