4) find the vertex, x - intercepts, and y - intercept of the graph of: $f(x)=x^{2}-2x - 8$ and then graph…

4) find the vertex, x - intercepts, and y - intercept of the graph of: $f(x)=x^{2}-2x - 8$ and then graph $f(x)$.

4) find the vertex, x - intercepts, and y - intercept of the graph of: $f(x)=x^{2}-2x - 8$ and then graph $f(x)$.

Answer

Explanation:

Step1: Find the vertex

For a quadratic function (y = ax^{2}+bx + c), the (x) - coordinate of the vertex is given by (x=-\frac{b}{2a}). For (f(x)=x^{2}-2x - 8), where (a = 1), (b=-2), (c=-8). (x=-\frac{-2}{2\times1}=1) Substitute (x = 1) into (f(x)): (y=f(1)=1^{2}-2\times1 - 8=1 - 2-8=-9) So the vertex is ((1,-9))

Step2: Find the (x) - intercepts

Set (y = 0), so (x^{2}-2x - 8=0) Factor the quadratic equation: (x^{2}-2x - 8=(x - 4)(x + 2)=0) Using the zero - product property (x-4=0) or (x + 2=0) (x=4) or (x=-2) The (x) - intercepts are ((4,0)) and ((-2,0))

Step3: Find the (y) - intercept

Set (x = 0), then (y=f(0)=0^{2}-2\times0 - 8=-8) The (y) - intercept is ((0,-8))

Answer:

  • Vertex: ((1,-9))
  • (x) - intercepts: ((4,0)) and ((-2,0))
  • (y) - intercept: ((0,-8))