Math  /  Algebra

QuestionFor the quadratic function h(x)=3x2+18x8h(x)=-3 x^{2}+18 x-8 : a) Write the equation in vertex form by completing square (2 marks) b) Write the coordinates of the vertex. (1 mark) c) Determine the domain of h(x)h(x). (1 mark) f) Determine the range of h(x)h(x). (1 mark) g) Graph the function. (3 marks) \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline & & & 4 \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. The given quadratic function is h(x)=3x2+18x8 h(x) = -3x^2 + 18x - 8 .
2. We need to convert the quadratic function into vertex form by completing the square.
3. The vertex form of a quadratic function is h(x)=a(xh)2+k h(x) = a(x-h)^2 + k , where (h,k)(h, k) is the vertex.
4. The domain of any quadratic function is all real numbers.
5. The range of a quadratic function depends on the direction of the parabola (upward or downward).

STEP 2

1. Convert the quadratic function to vertex form by completing the square.
2. Identify the vertex from the vertex form.
3. Determine the domain of the function.
4. Determine the range of the function.
5. Graph the function.

STEP 3

Start with the quadratic function:
h(x)=3x2+18x8 h(x) = -3x^2 + 18x - 8
Factor out the coefficient of x2x^2 from the first two terms:
h(x)=3(x26x)8 h(x) = -3(x^2 - 6x) - 8
Complete the square inside the parentheses. Take half of the coefficient of xx, square it, and add and subtract it inside the parentheses:
h(x)=3(x26x+99)8 h(x) = -3(x^2 - 6x + 9 - 9) - 8
Rewrite the expression:
h(x)=3((x3)29)8 h(x) = -3((x - 3)^2 - 9) - 8
Distribute the 3-3:
h(x)=3(x3)2+278 h(x) = -3(x - 3)^2 + 27 - 8
Simplify:
h(x)=3(x3)2+19 h(x) = -3(x - 3)^2 + 19

STEP 4

The vertex form of the function is:
h(x)=3(x3)2+19 h(x) = -3(x - 3)^2 + 19
The vertex (h,k)(h, k) is:
(3,19) (3, 19)

STEP 5

The domain of any quadratic function is all real numbers:
Domain: (,) \text{Domain: } (-\infty, \infty)

STEP 6

Since the parabola opens downwards (as indicated by the negative coefficient of (x3)2(x - 3)^2), the range is:
Range: (,19] \text{Range: } (-\infty, 19]

STEP 7

To graph the function, plot the vertex (3,19)(3, 19) and sketch the parabola opening downwards. Use additional points if necessary to ensure accuracy.

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