Question

graph the quadratic function below.
y = 8(x - 4)(x - 5)

Explanation:

Step1: Find x - intercepts

Set \( y = 0 \), then \( 0=8(x - 4)(x - 5) \). Since \( 8
eq0 \), we solve \( (x - 4)(x - 5)=0 \). So \( x - 4 = 0 \) or \( x - 5 = 0 \), which gives \( x = 4 \) and \( x = 5 \). The x - intercepts are \( (4,0) \) and \( (5,0) \).

Step2: Find the vertex

First, expand the function: \( y=8(x^{2}-5x - 4x + 20)=8(x^{2}-9x + 20)=8x^{2}-72x + 160 \). For a quadratic function \( y = ax^{2}+bx + c \), the x - coordinate of the vertex is \( x=-\frac{b}{2a} \). Here, \( a = 8 \), \( b=-72 \), so \( x =-\frac{-72}{2\times8}=\frac{72}{16}=\frac{9}{2}=4.5 \). Substitute \( x = 4.5 \) into the function: \( y=8(4.5 - 4)(4.5 - 5)=8(0.5)(-0.5)=8\times(-0.25)=-2 \). The vertex is \( (4.5,-2) \).

Step3: Determine the direction of the parabola

Since \( a = 8>0 \), the parabola opens upwards. Now, plot the x - intercepts \( (4,0) \), \( (5,0) \) and the vertex \( (4.5,-2) \), then draw a parabola opening upwards passing through these points.

(Note: If we were to present the graph description in a more formal way for plotting, we can also find additional points. For example, when \( x = 3 \), \( y=8(3 - 4)(3 - 5)=8(-1)(-2)=16 \), so the point \( (3,16) \); when \( x = 6 \), \( y=8(6 - 4)(6 - 5)=8(2)(1)=16 \), so the point \( (6,16) \). These points can help in drawing a more accurate parabola.)

Answer:

To graph \( y = 8(x - 4)(x - 5) \):

1. Plot the x - intercepts \( (4,0) \) and \( (5,0) \).
2. Plot the vertex \( (4.5,-2) \).
3. Since \( a = 8>0 \), draw a parabola opening upwards passing through these points (and additional points like \( (3,16) \), \( (6,16) \) for accuracy).