Graph f(x)=(x-2)^2-1

Problem

ƒ(x)=(x−2)2−1

Solution

1. Identify the form of the quadratic function. The function is given in vertex form, ƒ(x)=a*(x−h)2+k where (h,k) is the vertex.

2. Determine the vertex. By comparing ƒ(x)=(x−2)2−1 to the vertex form, we find h=2 and k=−1 The vertex is (2,−1)

3. Find the y-intercept. Set x=0 and evaluate the function.

ƒ(0)=(0−2)2−1

ƒ(0)=4−1

ƒ(0)=3

The y-intercept is (0,3)
4. Find the x-intercepts. Set ƒ(x)=0 and solve for x

(x−2)2−1=0

(x−2)2=1

x−2=±1

x=2±1

The x-intercepts are (1,0) and (3,0)
5. Determine the direction of the parabola. Since the leading coefficient a=1 is positive, the parabola opens upward.
6. Plot the points (2,−1) (0,3) (1,0) and (3,0) and draw a smooth curve through them to form the parabola.

Final Answer

ƒ(x)=(x−2)2−1

The graph is an upward-opening parabola with vertex (2,−1) y-intercept (0,3) and x-intercepts (1,0) and (3,0)

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