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.
f(x)equals4 left parenthesis x minus 2 right parenthesis squared minus 2
Question content area bottom left
Part 1
Use the graphing tool to graph the equation. Use the vertex and the y-intercept when drawing the graph.
.
.
.
Question content area right
Part 1
-10
-8
-6
-4
-2
2
4
6
8
10
-8
-4
4
8
12
16
20
x
y
f(x)equals4 left parenthesis x minus 2 right parenthesis squared minus 2
Question content area bottom left
Part 1
Use the graphing tool to graph the equation. Use the vertex and the y-intercept when drawing the graph.
.
.
.
Question content area right
Part 1
-10
-8
-6
-4
-2
2
4
6
8
10
-8
-4
4
8
12
16
20
x
y
Answers
GPT-4o mini
To analyze the quadratic function \( f(x) = 4(x - 2)^2 - 2 \), we will identify its vertex, intercepts, axis of symmetry, domain, and range.
### Step 1: Identify the Vertex
The function is in the vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. In this case:
- \( h = 2 \)
- \( k = -2 \)
Thus, the vertex is \( (2, -2) \).
### Step 2: Find the Y-Intercept
To find the y-intercept, we set \( x = 0 \):
\[
f(0) = 4(0 - 2)^2 - 2 = 4(4) - 2 = 16 - 2 = 14.
\]
So, the y-intercept is \( (0, 14) \).
### Step 3: Find the X-Intercepts
To find the x-intercepts, set \( f(x) = 0 \):
\[
4(x - 2)^2 - 2 = 0
\]
\[
4(x - 2)^2 = 2
\]
\[
(x - 2)^2 = \frac{1}{2}
\]
Taking the square root of both sides:
\[
x - 2 = \pm \sqrt{\frac{1}{2}}
\]
Thus, we have:
\[
x = 2 \pm \frac{\sqrt{2}}{2}.
\]
Calculating approximate values:
- \( x_1 \approx 2 + 0.707 = 2.707 \)
- \( x_2 \approx 2 - 0.707 = 1.293 \)
The x-intercepts are approximately \( (2.707, 0) \) and \( (1.293, 0) \).
### Step 4: Find the Axis of Symmetry
The axis of symmetry of the parabola is the vertical line that goes through the vertex:
\[
x = 2.
\]
### Step 5: Determine the Domain and Range
- **Domain**: The domain of any quadratic function is all real numbers, \( (-\infty, \infty) \).
- **Range**: Since the parabola opens upwards (because \( a = 4 > 0 \)), and the vertex is the minimum point at \( y = -2 \), the range is:
\[
[-2, \infty).
\]
### Summary
- **Vertex**: \( (2, -2) \)
- **Y-Intercept**: \( (0, 14) \)
- **X-Intercepts**: Approx. \( (2.707, 0) \) and \( (1.293, 0) \)
- **Axis of Symmetry**: \( x = 2 \)
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( [-2, \infty) \)
### Sketching the Graph
To sketch the graph:
1. Mark the vertex at \( (2, -2) \).
2. Plot the y-intercept at \( (0, 14) \) and the x-intercepts at approximately \( (2.707, 0) \) and \( (1.293, 0) \).
3. Draw a parabola opening upwards through these points.
Using graphing software or graph paper, you can now draw the graph reflecting the information above, checking that it opens upward and is symmetric about the line \( x = 2 \).
### Step 1: Identify the Vertex
The function is in the vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. In this case:
- \( h = 2 \)
- \( k = -2 \)
Thus, the vertex is \( (2, -2) \).
### Step 2: Find the Y-Intercept
To find the y-intercept, we set \( x = 0 \):
\[
f(0) = 4(0 - 2)^2 - 2 = 4(4) - 2 = 16 - 2 = 14.
\]
So, the y-intercept is \( (0, 14) \).
### Step 3: Find the X-Intercepts
To find the x-intercepts, set \( f(x) = 0 \):
\[
4(x - 2)^2 - 2 = 0
\]
\[
4(x - 2)^2 = 2
\]
\[
(x - 2)^2 = \frac{1}{2}
\]
Taking the square root of both sides:
\[
x - 2 = \pm \sqrt{\frac{1}{2}}
\]
Thus, we have:
\[
x = 2 \pm \frac{\sqrt{2}}{2}.
\]
Calculating approximate values:
- \( x_1 \approx 2 + 0.707 = 2.707 \)
- \( x_2 \approx 2 - 0.707 = 1.293 \)
The x-intercepts are approximately \( (2.707, 0) \) and \( (1.293, 0) \).
### Step 4: Find the Axis of Symmetry
The axis of symmetry of the parabola is the vertical line that goes through the vertex:
\[
x = 2.
\]
### Step 5: Determine the Domain and Range
- **Domain**: The domain of any quadratic function is all real numbers, \( (-\infty, \infty) \).
- **Range**: Since the parabola opens upwards (because \( a = 4 > 0 \)), and the vertex is the minimum point at \( y = -2 \), the range is:
\[
[-2, \infty).
\]
### Summary
- **Vertex**: \( (2, -2) \)
- **Y-Intercept**: \( (0, 14) \)
- **X-Intercepts**: Approx. \( (2.707, 0) \) and \( (1.293, 0) \)
- **Axis of Symmetry**: \( x = 2 \)
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( [-2, \infty) \)
### Sketching the Graph
To sketch the graph:
1. Mark the vertex at \( (2, -2) \).
2. Plot the y-intercept at \( (0, 14) \) and the x-intercepts at approximately \( (2.707, 0) \) and \( (1.293, 0) \).
3. Draw a parabola opening upwards through these points.
Using graphing software or graph paper, you can now draw the graph reflecting the information above, checking that it opens upward and is symmetric about the line \( x = 2 \).