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)=4-(x-3)^2
Use the graphing tool to graph the equation. Use the
vertex and one of the intercepts when drawing the
graph.
The axis of symmetry is blank.
(Type an equation. Simplify your answer.)
Identify the function's domain.
The domain is blank.
(Type the answer in interval notation.)
Identify the function's range.
The range is bank.
​(Type the answer in interval​ notation.)
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)=4-(x-3)^2
Use the graphing tool to graph the equation. Use the
vertex and one of the intercepts when drawing the
graph.
The axis of symmetry is blank.
(Type an equation. Simplify your answer.)
Identify the function's domain.
The domain is blank.
(Type the answer in interval notation.)
Identify the function's range.
The range is bank.
​(Type the answer in interval​ notation.)
Answers
GPT-4o mini
To analyze the quadratic function \( f(x) = 4 - (x - 3)^2 \), let's find the vertex, intercepts, axis of symmetry, domain, and range.
### 1. **Vertex**
The function is in vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, we have:
- \( h = 3 \)
- \( k = 4 \)
Thus, the vertex is at the point \((3, 4)\).
### 2. **Intercepts**
- **y-intercept**: This is found by evaluating \( f(0) \):
\[
f(0) = 4 - (0 - 3)^2 = 4 - 9 = -5
\]
So the y-intercept is \( (0, -5) \).
- **x-intercepts**: Set \( f(x) = 0 \):
\[
0 = 4 - (x - 3)^2
\]
Rearranging gives:
\[
(x - 3)^2 = 4
\]
Taking the square root of both sides:
\[
x - 3 = 2 \quad \text{or} \quad x - 3 = -2
\]
Thus,
\[
x = 5 \quad \text{or} \quad x = 1
\]
So the x-intercepts are \( (1, 0) \) and \( (5, 0) \).
### 3. **Axis of Symmetry**
The axis of symmetry is the vertical line that passes through the vertex:
\[
x = 3
\]
### 4. **Domain**
For quadratic functions, the domain is all real numbers:
\[
\text{Domain: } (-\infty, \infty)
\]
### 5. **Range**
Since the parabola opens downwards (the coefficient of \( (x-3)^2 \) is negative), the range will be all values less than or equal to the y-coordinate of the vertex:
\[
\text{Range: } (-\infty, 4]
\]
### Summary
- **Axis of symmetry**: \( x = 3 \)
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (-\infty, 4] \)
Now you can sketch the parabola using the vertex (3, 4) and the intercepts (0, -5), (1, 0), and (5, 0). The graph will be a downward-opening parabola.
### 1. **Vertex**
The function is in vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, we have:
- \( h = 3 \)
- \( k = 4 \)
Thus, the vertex is at the point \((3, 4)\).
### 2. **Intercepts**
- **y-intercept**: This is found by evaluating \( f(0) \):
\[
f(0) = 4 - (0 - 3)^2 = 4 - 9 = -5
\]
So the y-intercept is \( (0, -5) \).
- **x-intercepts**: Set \( f(x) = 0 \):
\[
0 = 4 - (x - 3)^2
\]
Rearranging gives:
\[
(x - 3)^2 = 4
\]
Taking the square root of both sides:
\[
x - 3 = 2 \quad \text{or} \quad x - 3 = -2
\]
Thus,
\[
x = 5 \quad \text{or} \quad x = 1
\]
So the x-intercepts are \( (1, 0) \) and \( (5, 0) \).
### 3. **Axis of Symmetry**
The axis of symmetry is the vertical line that passes through the vertex:
\[
x = 3
\]
### 4. **Domain**
For quadratic functions, the domain is all real numbers:
\[
\text{Domain: } (-\infty, \infty)
\]
### 5. **Range**
Since the parabola opens downwards (the coefficient of \( (x-3)^2 \) is negative), the range will be all values less than or equal to the y-coordinate of the vertex:
\[
\text{Range: } (-\infty, 4]
\]
### Summary
- **Axis of symmetry**: \( x = 3 \)
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (-\infty, 4] \)
Now you can sketch the parabola using the vertex (3, 4) and the intercepts (0, -5), (1, 0), and (5, 0). The graph will be a downward-opening parabola.