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Math  /  Algebra

Questiony=−(x−3)(x−3)y=-(x-3)(x-3) x-intercepts: ( □\square , 0) □\square ,0)
Axis of Symmetry: □\square
Vertex: □\square y-intercept: (0, □\square

Studdy Solution

STEP 1

1. The given function is a quadratic function in the form y=a(x−h)2+k y = a(x-h)^2 + k .
2. The function is already factored as y=−(x−3)(x−3) y = -(x-3)(x-3) , which can be rewritten as y=−(x−3)2 y = -(x-3)^2 .
3. The x-intercepts occur where y=0 y = 0 .
4. The axis of symmetry for a parabola in vertex form y=a(x−h)2+k y = a(x-h)^2 + k is x=h x = h .
5. The vertex of the parabola is at the point (h,k) (h, k) .
6. The y-intercept occurs where x=0 x = 0 .

STEP 2

1. Identify the form of the function and rewrite if necessary.
2. Find the x-intercepts.
3. Determine the axis of symmetry.
4. Identify the vertex.
5. Calculate the y-intercept.

STEP 3

The given function is y=−(x−3)(x−3) y = -(x-3)(x-3) . This can be rewritten as:
y=−(x−3)2 y = -(x-3)^2
This is the vertex form of a quadratic function, where a=−1 a = -1 , h=3 h = 3 , and k=0 k = 0 .

STEP 4

To find the x-intercepts, set y=0 y = 0 and solve for x x :
0=−(x−3)2 0 = -(x-3)^2
Divide both sides by −1-1:
0=(x−3)2 0 = (x-3)^2
Take the square root of both sides:
x−3=0 x - 3 = 0
Solve for x x :
x=3 x = 3
Thus, the x-intercept is at (3,0) (3, 0) .

STEP 5

The axis of symmetry for the function y=−(x−3)2 y = -(x-3)^2 is given by x=h x = h , where h=3 h = 3 .
Thus, the axis of symmetry is x=3 x = 3 .

STEP 6

The vertex of the parabola in the form y=a(x−h)2+k y = a(x-h)^2 + k is at the point (h,k) (h, k) .
For the function y=−(x−3)2 y = -(x-3)^2 , the vertex is at (3,0) (3, 0) .

STEP 7

To find the y-intercept, set x=0 x = 0 and solve for y y :
y=−(0−3)2 y = -(0-3)^2 y=−(3)2 y = -(3)^2 y=−9 y = -9
Thus, the y-intercept is at (0,−9) (0, -9) .
The x-intercepts are (3,0) (3, 0) , the axis of symmetry is x=3 x = 3 , the vertex is (3,0) (3, 0) , and the y-intercept is (0,−9) (0, -9) .

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