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 left parenthesis x right parenthesis equals 9 minus left parenthesis x minus 4 right parenthesis squared
Question content area bottom left
Part 1
Use the graphing tool to graph the equation. Use the vertex and one of the intercepts when drawing the graph.
.
.
.
Question content area right
Part 1
-10
-8
-6
-4
-2
2
4
6
8
10
-10
-8
-6
-4
-2
2
4
6
8
10
x
y
f left parenthesis x right parenthesis equals 9 minus left parenthesis x minus 4 right parenthesis squared
Question content area bottom left
Part 1
Use the graphing tool to graph the equation. Use the vertex and one of the intercepts when drawing the graph.
.
.
.
Question content area right
Part 1
-10
-8
-6
-4
-2
2
4
6
8
10
-10
-8
-6
-4
-2
2
4
6
8
10
x
y
Answers
GPT-4o mini
To analyze the quadratic function \( f(x) = 9 - (x - 4)^2 \), let's start by identifying key features such as the vertex, intercepts, axis of symmetry, domain, and range.
### Step 1: Vertex
The function is in vertex form, which is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. In this case, we can see:
- \( h = 4 \)
- \( k = 9 \)
Thus, the vertex is at the point \( (4, 9) \).
### Step 2: Axis of Symmetry
The axis of symmetry can be found using the vertex \( x = h \). Therefore:
\[
\text{Axis of symmetry: } x = 4
\]
### Step 3: Finding the Intercepts
#### y-intercept
To find the y-intercept, set \( x = 0 \):
\[
f(0) = 9 - (0 - 4)^2 = 9 - 16 = -7
\]
Thus, the y-intercept is \( (0, -7) \).
#### x-intercepts
To find the x-intercepts, set \( f(x) = 0 \):
\[
0 = 9 - (x - 4)^2
\]
\[
(x - 4)^2 = 9
\]
Taking the square root of both sides:
\[
x - 4 = 3 \quad \text{or} \quad x - 4 = -3
\]
Solving these gives:
\[
x = 7 \quad \text{or} \quad x = 1
\]
Thus, the x-intercepts are \( (7, 0) \) and \( (1, 0) \).
### Step 4: Domain and Range
**Domain:**
The domain of any quadratic function is all real numbers:
\[
\text{Domain: } (-\infty, \infty)
\]
**Range:**
Since the vertex is at \( (4, 9) \) and the coefficient of the squared term is negative (indicating the parabola opens downward), the maximum value of \( f(x) \) is \( 9 \) (the y-coordinate of the vertex). Therefore, the range is:
\[
\text{Range: } (-\infty, 9]
\]
### Summary
- **Vertex:** \( (4, 9) \)
- **Axis of symmetry:** \( x = 4 \)
- **y-intercept:** \( (0, -7) \)
- **x-intercepts:** \( (1, 0) \) and \( (7, 0) \)
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( (-\infty, 9] \)
### Sketching the Graph
Here’s how to sketch the graph based on the information we have:
1. Plot the vertex at \( (4, 9) \).
2. Plot the y-intercept at \( (0, -7) \).
3. Plot the x-intercepts at \( (1, 0) \) and \( (7, 0) \).
4. Draw a smooth curve through these points, opening downwards.
The parabola should reflect across the axis of symmetry \( x = 4 \).
You can use graphing tools or graph paper to sketch the parabola accurately for a visual representation.
### Step 1: Vertex
The function is in vertex form, which is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. In this case, we can see:
- \( h = 4 \)
- \( k = 9 \)
Thus, the vertex is at the point \( (4, 9) \).
### Step 2: Axis of Symmetry
The axis of symmetry can be found using the vertex \( x = h \). Therefore:
\[
\text{Axis of symmetry: } x = 4
\]
### Step 3: Finding the Intercepts
#### y-intercept
To find the y-intercept, set \( x = 0 \):
\[
f(0) = 9 - (0 - 4)^2 = 9 - 16 = -7
\]
Thus, the y-intercept is \( (0, -7) \).
#### x-intercepts
To find the x-intercepts, set \( f(x) = 0 \):
\[
0 = 9 - (x - 4)^2
\]
\[
(x - 4)^2 = 9
\]
Taking the square root of both sides:
\[
x - 4 = 3 \quad \text{or} \quad x - 4 = -3
\]
Solving these gives:
\[
x = 7 \quad \text{or} \quad x = 1
\]
Thus, the x-intercepts are \( (7, 0) \) and \( (1, 0) \).
### Step 4: Domain and Range
**Domain:**
The domain of any quadratic function is all real numbers:
\[
\text{Domain: } (-\infty, \infty)
\]
**Range:**
Since the vertex is at \( (4, 9) \) and the coefficient of the squared term is negative (indicating the parabola opens downward), the maximum value of \( f(x) \) is \( 9 \) (the y-coordinate of the vertex). Therefore, the range is:
\[
\text{Range: } (-\infty, 9]
\]
### Summary
- **Vertex:** \( (4, 9) \)
- **Axis of symmetry:** \( x = 4 \)
- **y-intercept:** \( (0, -7) \)
- **x-intercepts:** \( (1, 0) \) and \( (7, 0) \)
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( (-\infty, 9] \)
### Sketching the Graph
Here’s how to sketch the graph based on the information we have:
1. Plot the vertex at \( (4, 9) \).
2. Plot the y-intercept at \( (0, -7) \).
3. Plot the x-intercepts at \( (1, 0) \) and \( (7, 0) \).
4. Draw a smooth curve through these points, opening downwards.
The parabola should reflect across the axis of symmetry \( x = 4 \).
You can use graphing tools or graph paper to sketch the parabola accurately for a visual representation.