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Math  /  Algebra

QuestionUse the vertex and intercepts to sketch the graph of the quadratic function. Give the equation for the parabola's axis of symmetry. Use the parabola to identify the function's domain and range. f(x)=(x−2)2+4f(x)=(x-2)^{2}+4
Use the graphing tool to graph the equation. Use the vertex and the yy-intercept when drawing the graph.
The axis of symmetry is □\square (Simplify your answer. Type an equation.) Identify the function's domain. The domain is □\square (Type the answer in interval notation.) Identify the function's range. The range is □\square (Type the answer in interval notation.)

Studdy Solution

STEP 1

1. The function f(x)=(x−2)2+4 f(x) = (x-2)^2 + 4 is a quadratic function in vertex form.
2. The vertex form of a quadratic function is f(x)=a(x−h)2+k f(x) = a(x-h)^2 + k , where (h,k)(h, k) is the vertex.
3. The axis of symmetry is a vertical line that passes through the vertex.
4. The domain of any quadratic function is all real numbers.
5. The range of the quadratic function depends on whether it opens upwards or downwards.

STEP 2

1. Identify the vertex of the parabola.
2. Determine the axis of symmetry.
3. Identify the y-intercept.
4. Determine the domain of the function.
5. Determine the range of the function.

STEP 3

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

STEP 4

Determine the axis of symmetry. The axis of symmetry for a parabola in vertex form is x=h x = h .
Thus, the axis of symmetry is:
x=2 x = 2

STEP 5

Identify the y-intercept. The y-intercept occurs when x=0 x = 0 .
Substitute x=0 x = 0 into the function:
f(0)=(0−2)2+4=4+4=8 f(0) = (0-2)^2 + 4 = 4 + 4 = 8
The y-intercept is (0,8)(0, 8).

STEP 6

Determine the domain of the function. Since this is a quadratic function, the domain is all real numbers.
The domain is:
(−∞,∞) (-\infty, \infty)

STEP 7

Determine the range of the function. Since the parabola opens upwards (the coefficient of (x−2)2(x-2)^2 is positive), the range starts at the y-coordinate of the vertex and goes to infinity.
The range is:
[4,∞) [4, \infty)
The axis of symmetry is x=2 x = 2 .
The domain is (−∞,∞) (-\infty, \infty) .
The range is [4,∞) [4, \infty) .

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