Part 1 of 4
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.

Question

UpStudy AI · Accepted Answer

**Step 1. Identify the vertex**

The quadratic function is given by  
$$
f(x)=(x-2)^2+1.
$$  
In vertex form $$ f(x) = a(x-h)^2 + k $$, the vertex is at $$(h,k)$$. Here,  
$$
h=2 \quad \text{and} \quad k=1,
$$  
so the vertex is at $$(2, 1)$$.

**Step 2. Determine the axis of symmetry**

The axis of symmetry of a parabola in vertex form is the vertical line through the vertex, which is given by  
$$
x = h.
$$  
Thus, for our function, the axis of symmetry is  
$$
x=2.
$$

**Step 3. Find the intercepts**

***Y-intercept:***  
Set $$ x=0 $$ in the function:
$$
f(0) = (0-2)^2 + 1 = 4 + 1 = 5.
$$  
So, the y-intercept is at $$(0, 5)$$.

***X-intercepts:***  
Set $$ f(x)=0 $$ to find the x-intercepts:
$$
(x-2)^2+1=0 \quad \Rightarrow \quad (x-2)^2=-1.
$$  
Since $$(x-2)^2 \ge 0$$ for all real $$x$$, there is no real solution. Therefore, there are no x-intercepts.

**Step 4. Determine the domain and range**

***Domain:***  
The function $$ f(x) $$ is defined for all real numbers. Thus, the domain is  
$$
(-\infty,\infty).
$$

***Range:***  
Since the quadratic has a minimum at the vertex and opens upward (because the coefficient of $$(x-2)^2$$ is positive), the range is all $$ y $$ values greater than or equal to the y-coordinate of the vertex. Therefore, the range is  
$$
[1,\infty).
$$

**Summary of Answers**

- **Vertex:** $$(2,1)$$
- **Axis of symmetry:** $$\displaystyle x=2$$
- **Y-intercept:** $$(0,5)$$
- **X-intercepts:** None (no real x-intercepts)
- **Domain:** $$(-\infty,\infty)$$
- **Range:** $$[1,\infty)$$
**Summary:** - **Vertex:** $$(2, 1)$$ - **Axis of Symmetry:** $$x = 2$$ - **Y-intercept:** $$(0, 5)$$ - **X-intercepts:** None - **Domain:** All real numbers - **Range:** $$y \geq 1$$

Part 1 of 4
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.

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Latest Algebra Questions

Part 1 of 4 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.

Upstudy AI Solution

Answer

Solution

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Latest Algebra Questions

Part 1 of 4
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.