Question

find the x-intercept(s) and the coordinates of the vertex for the parabola ( y = x^2 - 8x + 7 ). if there is more than one x-intercept, separate them with commas.
x-intercept(s):
vertex: ( , )

Explanation:

Step1: Find x-intercepts

Set \( y = 0 \) in \( y = x^2 - 8x + 7 \). So, \( x^2 - 8x + 7 = 0 \). Factor the quadratic: \( (x - 1)(x - 7)=0 \). Solve for \( x \): \( x - 1 = 0 \) or \( x - 7 = 0 \), so \( x = 1 \) or \( x = 7 \).

Step2: Find vertex coordinates

For a quadratic \( y = ax^2 + bx + c \), the x - coordinate of the vertex is \( x = -\frac{b}{2a} \). Here, \( a = 1 \), \( b = -8 \), so \( x = -\frac{-8}{2\times1}=4 \). Substitute \( x = 4 \) into the equation: \( y = 4^2 - 8\times4 + 7 = 16 - 32 + 7=-9 \). So the vertex is \( (4, -9) \).

Answer:

x - intercept(s): \( 1, 7 \)
vertex: \( (4, -9) \)