5. Determine the $x$-intercepts and the vertex of the parabola. Show your work. Then graph the parabola.
\[
y=x^{2}-8 x+15
\]

Answer

Expert–verified

Hide Steps

Answer

$\boxed{\text{The x-intercepts are 3 and 5, and the vertex of the parabola is (4, -1)}}$

Steps

Step 1 ：Set y to 0 and solve for x: $0 = x^2 - 8x + 15$

Step 2 ：Factor the equation: $(x - 3)(x - 5) = 0$

Step 3 ：Find the x-intercepts: $x = 3$ and $x = 5$

Step 4 ：Use the vertex formula $x = \frac{-b}{2a}$ with a = 1 and b = -8: $x = \frac{-(-8)}{2(1)} = 4$

Step 5 ：Plug the x-coordinate of the vertex back into the equation to find the y-coordinate: $y = (4)^2 - 8(4) + 15 = -1$

Step 6 ：$\boxed{\text{The x-intercepts are 3 and 5, and the vertex of the parabola is (4, -1)}}$