Question

question 6 of 10
which graph is defined by the function given below?
y = (x - 2)(x + 5)
four graphs labeled a, b, c, d
click here for long description
a. graph a
b. graph b
c. graph c
d. graph d

Explanation:

Step1: Find x - intercepts

To find the x - intercepts, set \(y = 0\). So, \(0=(x - 2)(x + 5)\). Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So, \(x-2=0\) gives \(x = 2\) and \(x + 5=0\) gives \(x=-5\). The x - intercepts of the parabola are at \(x=-5\) and \(x = 2\).

Step2: Analyze the parabola's direction

The function is in the form \(y=(x - 2)(x + 5)=x^{2}+3x - 10\). For a quadratic function of the form \(y = ax^{2}+bx + c\), when \(a>0\) (here \(a = 1>0\)), the parabola opens upwards.

Now, let's analyze the x - intercepts. The x - intercepts are at \(x=-5\) and \(x = 2\). Let's check the graphs:

• Graph A: Let's assume the x - intercepts. If we look at the x - intercepts, for the function \(y=(x - 2)(x + 5)\), the roots are at \(x=-5\) and \(x = 2\). Let's check the y - intercept. When \(x = 0\), \(y=(0 - 2)(0 + 5)=- 10\). So the y - intercept is \((0,-10)\).
• Graph B: The x - intercepts do not match \(x=-5\) and \(x = 2\).
• Graph C: The x - intercepts do not match \(x=-5\) and \(x = 2\) (it seems to be symmetric about the y - axis, but our function is not symmetric about the y - axis since the axis of symmetry is \(x=-\frac{b}{2a}=-\frac{3}{2}

eq0\)).

• Graph D: The x - intercepts do not match \(x=-5\) and \(x = 2\).

The graph with x - intercepts at \(x=-5\) and \(x = 2\), opening upwards and y - intercept at \((0,-10)\) is Graph A.

Answer:

A. Graph A