Math

QuestionSketch the graph of f(x)=(x2)2+4f(x)=(x-2)^{2}+4. Find the axis of symmetry, domain, and range in interval notation.

Studdy Solution

STEP 1

Assumptions1. The function is a quadratic function given by f(x)=(x)+4f(x)=(x-)^{}+4 . The vertex form of a quadratic function is f(x)=a(xh)+kf(x)=a(x-h)^{}+k, where (h,k)(h,k) is the vertex of the parabola3. The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola4. The domain of a function is the set of all possible x-values, and the range is the set of all possible y-values5. For a parabola that opens upwards or downwards, the domain is all real numbers, and the range is either [k,)[k, \infty) or (,k](-\infty, k], where k is the y-coordinate of the vertex

STEP 2

First, identify the vertex of the parabola. The vertex is given by the point (h,k)(h,k) in the vertex form of the quadratic function.
Vertex=(h,k)Vertex = (h, k)

STEP 3

Now, plug in the given values for h and k to find the vertex.
Vertex=(2,)Vertex = (2,)

STEP 4

Next, identify the y-intercept of the parabola. The y-intercept is the point where the parabola crosses the y-axis. This occurs when x=0x=0.
yintercept=f(0)y-intercept = f(0)

STEP 5

Now, plug in x=0x=0 into the function to find the y-intercept.
yintercept=f(0)=(02)2+4y-intercept = f(0) = (0-2)^{2}+4

STEP 6

Calculate the y-intercept.
yintercept=4+4=8y-intercept =4 +4 =8

STEP 7

The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry is x=hx=h, where h is the x-coordinate of the vertex.
Axisofsymmetry=x=hAxis\, of\, symmetry = x = h

STEP 8

Now, plug in the x-coordinate of the vertex to find the equation of the axis of symmetry.
Axisofsymmetry=x=2Axis\, of\, symmetry = x =2

STEP 9

The domain of a function is the set of all possible x-values. For a parabola, the domain is all real numbers.
Domain=(,)Domain = (-\infty, \infty)

STEP 10

The range of a function is the set of all possible y-values. For a parabola that opens upwards, the range is [k,)[k, \infty), where k is the y-coordinate of the vertex.
Range=[k,)Range = [k, \infty)

STEP 11

Now, plug in the y-coordinate of the vertex to find the range.
Range=[4,)Range = [4, \infty)The vertex of the parabola is (,4)(,4), the y-intercept is8, the axis of symmetry is x=x=, the domain is all real numbers, and the range is [4,)[4, \infty).

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