Use 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.

Use the graphing tool to graph the equation. Use the vertex and the y-intercept when
drawing the graph.
The axis of symmetry is
(Simplify your answer. Types an equation.)

Question

Use 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.

Use the graphing tool to graph the equation. Use the vertex and the y-intercept when
drawing the graph.
The axis of symmetry is
(Simplify your answer. Types an equation.)

UpStudy AI · Accepted Answer

1. We start with the quadratic function  
   $$
   f(x)=(x-2)^2+1.
   $$

2. The function is written in vertex form  
   $$
   f(x)=a(x-h)^2+k,
   $$  
   where the vertex is at $$(h,k)$$. Here,  
   $$
   h=2 \quad \text{and} \quad k=1.
   $$  
   Thus, the vertex is  
   $$
   (2,1).
   $$

3. The axis of symmetry for a parabola in vertex form is the vertical line passing through the vertex. Therefore, the axis of symmetry is  
   $$
   x=2.
   $$

4. To find the $$y$$-intercept, set $$x=0$$:  
   $$
   f(0)=(0-2)^2+1=4+1=5.
   $$  
   So, the $$y$$-intercept is at  
   $$
   (0,5).
   $$

5. The quadratic function is a parabola that opens upward (since the coefficient of $$(x-2)^2$$ is positive). The vertex $$(2,1)$$ is therefore the minimum point of the parabola.

6. The domain of any quadratic function is all real numbers, so the domain is  
   $$
   (-\infty,\infty).
   $$

7. Since the vertex is the minimum point and the parabola opens upward, the range of the function is  
   $$
   [1,\infty).
   $$

8. In summary, the key features of the quadratic function are:  
   - **Vertex:** $$(2,1)$$  
   - **Axis of Symmetry:** $$\boxed{x=2}$$  
   - **$$y$$-Intercept:** $$(0,5)$$  
   - **Domain:** $$(-\infty,\infty)$$  
   - **Range:** $$[1,\infty)$$
The axis of symmetry is $$ x = 2 $$.

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