Complete the square for the quadratic function y = x2−10x + 9 put the answer in vertex form y=a(x−h)2+k then identify the vertex and axis of symmetry. (You are NOT Solving for x)(4 points)
To complete the square for the quadratic function y = x^2 - 10x + 9, we need to follow these steps:
Step 1: Take half of the coefficient of x, square it, and add it to both sides of the equation.
Step 2: Rewrite the quadratic trinomial as a perfect square trinomial.
Step 3: Simplify the equation and rewrite it in vertex form.
Let's compute these steps:
Step 1:
Take half of the coefficient of x: (-10)/2 = -5.
Square -5: (-5)^2 = 25.
Add 25 to both sides of the equation:
y + 25 = x^2 - 10x + 25 + 9
Simplifying: y + 25 = x^2 - 10x + 34
Step 2:
Rewrite the quadratic trinomial as a perfect square trinomial:
y + 25 = (x^2 - 10x + 25) + 9
Simplifying: y + 25 = (x - 5)^2 + 9
Step 3:
Simplify the equation and rewrite it in vertex form:
y + 25 - 9 = (x - 5)^2
y + 16 = (x - 5)^2
Now the equation is in vertex form: y = (x - 5)^2 - 16.
The vertex form of the quadratic function is y = (x - 5)^2 - 16.
The vertex is the point (h, k), so the vertex of this quadratic function is (5, -16).
The axis of symmetry is a vertical line through the vertex x = h, so the axis of symmetry is x = 5.