Math / Algebra

QuestionShort Answer
10. (3 points) Sketch the graph of $f(x)=-(x-4)^{2}+2$ , then state the domain and range of the function.

Studdy Solution

STEP 1

1. The function $f(x) = -(x-4)^2 + 2$ is a quadratic function.
2. The graph of $f(x)$ is a parabola that opens downwards because of the negative sign in front of the squared term.
3. We need to determine the domain and range of the function after sketching the graph.

STEP 2

1. Identify the vertex of the parabola.
2. Determine the direction in which the parabola opens.
3. Sketch the graph based on the vertex and direction.
4. State the domain of the function.
5. State the range of the function.

STEP 3

The function is in the vertex form $f(x) = a(x-h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.
For $f(x) = -(x-4)^2 + 2$ , the vertex is $(4, 2)$ .

STEP 4

The coefficient of the squared term is negative ( $-1$ ), indicating that the parabola opens downwards.

STEP 5

To sketch the graph, plot the vertex at $(4, 2)$ and draw a downward-opening parabola.
The axis of symmetry is the vertical line $x = 4$ .

STEP 6

The domain of any quadratic function is all real numbers.
Thus, the domain of $f(x)$ is $(-\infty, \infty)$ .

STEP 7

Since the parabola opens downwards and the vertex is at the maximum point, the range is all values less than or equal to the y-coordinate of the vertex.
Thus, the range of $f(x)$ is $(-\infty, 2]$ .
The domain of the function is $(-\infty, \infty)$ and the range is $(-\infty, 2]$ .

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QuestionShort Answer
10. (3 points) Sketch the graph of $f(x)=-(x-4)^{2}+2$ , then state the domain and range of the function.

Studdy Solution

STEP 1

1. The function $f(x) = -(x-4)^2 + 2$ is a quadratic function.
2. The graph of $f(x)$ is a parabola that opens downwards because of the negative sign in front of the squared term.
3. We need to determine the domain and range of the function after sketching the graph.

STEP 2

1. Identify the vertex of the parabola.
2. Determine the direction in which the parabola opens.
3. Sketch the graph based on the vertex and direction.
4. State the domain of the function.
5. State the range of the function.

STEP 3

The function is in the vertex form $f(x) = a(x-h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.
For $f(x) = -(x-4)^2 + 2$ , the vertex is $(4, 2)$ .

STEP 4

The coefficient of the squared term is negative ( $-1$ ), indicating that the parabola opens downwards.

STEP 5

To sketch the graph, plot the vertex at $(4, 2)$ and draw a downward-opening parabola.
The axis of symmetry is the vertical line $x = 4$ .

STEP 6

The domain of any quadratic function is all real numbers.
Thus, the domain of $f(x)$ is $(-\infty, \infty)$ .

STEP 7

Since the parabola opens downwards and the vertex is at the maximum point, the range is all values less than or equal to the y-coordinate of the vertex.
Thus, the range of $f(x)$ is $(-\infty, 2]$ .
The domain of the function is $(-\infty, \infty)$ and the range is $(-\infty, 2]$ .

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QuestionShort Answer10. (3 points) Sketch the graph of f(x)=−(x−4)2+2f(x)=-(x-4)^{2}+2f(x)=−(x−4)2+2, then state the domain and range of the function.

Studdy Solution

STEP 1

STEP 2

1. Identify the vertex of the parabola.2. Determine the direction in which the parabola opens.3. Sketch the graph based on the vertex and direction.4. State the domain of the function.5. State the range of the function.

STEP 3

The function is in the vertex form f(x)=a(x−h)2+k f(x) = a(x-h)^2 + k f(x)=a(x−h)2+k, where (h,k) (h, k) (h,k) is the vertex of the parabola.For f(x)=−(x−4)2+2 f(x) = -(x-4)^2 + 2 f(x)=−(x−4)2+2, the vertex is (4,2) (4, 2) (4,2).

STEP 4

The coefficient of the squared term is negative (−1 -1 −1), indicating that the parabola opens downwards.

STEP 5

To sketch the graph, plot the vertex at (4,2) (4, 2) (4,2) and draw a downward-opening parabola.The axis of symmetry is the vertical line x=4 x = 4 x=4.

STEP 6

The domain of any quadratic function is all real numbers.Thus, the domain of f(x) f(x) f(x) is (−∞,∞) (-\infty, \infty) (−∞,∞).

STEP 7

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QuestionShort Answer
10. (3 points) Sketch the graph of $f(x)=-(x-4)^{2}+2$ , then state the domain and range of the function.

1. Identify the vertex of the parabola.
2. Determine the direction in which the parabola opens.
3. Sketch the graph based on the vertex and direction.
4. State the domain of the function.
5. State the range of the function.

The function is in the vertex form $f(x) = a(x-h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.
For $f(x) = -(x-4)^2 + 2$ , the vertex is $(4, 2)$ .

The coefficient of the squared term is negative ( $-1$ ), indicating that the parabola opens downwards.

To sketch the graph, plot the vertex at $(4, 2)$ and draw a downward-opening parabola.
The axis of symmetry is the vertical line $x = 4$ .

The domain of any quadratic function is all real numbers.
Thus, the domain of $f(x)$ is $(-\infty, \infty)$ .