Math

Question Find the maximum value of the quadratic function y=3x2+12x9y = -3x^2 + 12x - 9. The vertex is at xvertex=b/2ax_{\text{vertex}} = -b/2a, and the maximum value is y=3y = 3.

Studdy Solution

STEP 1

Assumptions
1. The given function is a quadratic function in the form y=ax2+bx+c y = ax^2 + bx + c .
2. The coefficients of the function are a=3 a = -3 , b=12 b = 12 , and c=9 c = -9 .
3. The vertex form of a quadratic function is y=a(xh)2+k y = a(x - h)^2 + k , where (h,k) (h, k) is the vertex of the parabola.
4. The x-coordinate of the vertex xvertex x_{\text{vertex}} can be found using the formula xvertex=b2a x_{\text{vertex}} = -\frac{b}{2a} .
5. Since a<0 a < 0 , the parabola opens downwards, and the vertex represents the maximum point on the graph.

STEP 2

First, we need to find the x-coordinate of the vertex of the parabola using the formula xvertex=b2a x_{\text{vertex}} = -\frac{b}{2a} .

STEP 3

Now, plug in the given values for a a and b b to calculate xvertex x_{\text{vertex}} .
xvertex=122(3) x_{\text{vertex}} = -\frac{12}{2(-3)}

STEP 4

Simplify the expression to find the value of xvertex x_{\text{vertex}} .
xvertex=126 x_{\text{vertex}} = -\frac{12}{-6}

STEP 5

Calculate the x-coordinate of the vertex.
xvertex=2 x_{\text{vertex}} = 2

STEP 6

Now that we have the x-coordinate of the vertex, we can find the maximum value of the function by substituting xvertex x_{\text{vertex}} back into the original equation.
y=3(2)2+12(2)9 y = -3(2)^2 + 12(2) - 9

STEP 7

Calculate the value inside the parentheses.
y=3(4)+249 y = -3(4) + 24 - 9

STEP 8

Multiply the terms to simplify the equation.
y=12+249 y = -12 + 24 - 9

STEP 9

Add and subtract the numbers to find the maximum value of the function.
y=129 y = 12 - 9

STEP 10

Calculate the maximum value of the function.
y=3 y = 3
The maximum value of the function is 3.

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