Chapter2: Functions And Their Graphs
Section2.4: A Library Of Parent Functions
Problem 47E: During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate...
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![### Graphing a Step Function on a Given Interval
**Task:**
Graph the function on the interval \([-5, 1]\).
**Function:**
\[ f(x) = \left\lfloor x + 5 \right\rfloor \]
**Steps:**
1. **Understanding the Function:**
- \( f(x) = \left\lfloor x + 5 \right\rfloor \) is known as the floor function or greatest integer function.
- This function takes the value of \( x + 5 \) and rounds it down to the nearest integer.
2. **Interval Definition:**
- The function needs to be graphed from \( x = -5 \) to \( x = 1 \).
**Graph Explanation:**
1. **Axes:**
- The horizontal axis represents \( x \).
- The vertical axis represents \( f(x) \).
2. **Plotting the Function:**
- At \( x = -5 \), \[ f(-5) = \left\lfloor -5 + 5 \right\rfloor = \left\lfloor 0 \right\rfloor = 0 \]
- At \( x = -4.5 \), \[ f(-4.5) = \left\lfloor -4.5 + 5 \right\rfloor = \left\lfloor 0.5 \right\rfloor = 0 \]
- At \( x = -4 \), \[ f(-4) = \left\lfloor -4 + 5 \right\rfloor = \left\lfloor 1 \right\rfloor = 1 \]
- Continue calculating \( f(x) \) for each interval, and plot the values on the graph.
**Graph Detailing:**
- The graph is a step function with a series of horizontal lines.
- Each horizontal line segment spans from one integer value of \( x + 5 \) to just before the next integer.
- The graph ‘steps up’ at every point where \( x \) increases by 1.
**Example Calculation and Plot Points:**
- For \( x \in [-5, -4) \), \( f(x) = 0 \)
- For \( x \in [-4, -3) \), \( f(x) = 1 \)
- For \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ffd9792-2b35-4da7-9b44-f091980b5633%2Fb1afa74e-bc88-4414-b5a4-5bfb6c900b77%2Fnsf3rfp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Graphing a Step Function on a Given Interval
**Task:**
Graph the function on the interval \([-5, 1]\).
**Function:**
\[ f(x) = \left\lfloor x + 5 \right\rfloor \]
**Steps:**
1. **Understanding the Function:**
- \( f(x) = \left\lfloor x + 5 \right\rfloor \) is known as the floor function or greatest integer function.
- This function takes the value of \( x + 5 \) and rounds it down to the nearest integer.
2. **Interval Definition:**
- The function needs to be graphed from \( x = -5 \) to \( x = 1 \).
**Graph Explanation:**
1. **Axes:**
- The horizontal axis represents \( x \).
- The vertical axis represents \( f(x) \).
2. **Plotting the Function:**
- At \( x = -5 \), \[ f(-5) = \left\lfloor -5 + 5 \right\rfloor = \left\lfloor 0 \right\rfloor = 0 \]
- At \( x = -4.5 \), \[ f(-4.5) = \left\lfloor -4.5 + 5 \right\rfloor = \left\lfloor 0.5 \right\rfloor = 0 \]
- At \( x = -4 \), \[ f(-4) = \left\lfloor -4 + 5 \right\rfloor = \left\lfloor 1 \right\rfloor = 1 \]
- Continue calculating \( f(x) \) for each interval, and plot the values on the graph.
**Graph Detailing:**
- The graph is a step function with a series of horizontal lines.
- Each horizontal line segment spans from one integer value of \( x + 5 \) to just before the next integer.
- The graph ‘steps up’ at every point where \( x \) increases by 1.
**Example Calculation and Plot Points:**
- For \( x \in [-5, -4) \), \( f(x) = 0 \)
- For \( x \in [-4, -3) \), \( f(x) = 1 \)
- For \(
![**Graph the Function**
Consider the piecewise function \( m(x) \) defined as follows:
\[
m(x) =
\begin{cases}
-3 & \text{for } -3 < x < -2 \\
x & \text{for } -2 \le x < 2 \\
-x^2 + 4 & \text{for } x \ge 2
\end{cases}
\]
### Explanation of the Piecewise Function:
1. **First Piece:** \( m(x) = -3 \) for \( -3 < x < -2 \)
- This piece of the function is a constant, where the function value is -3 for all \( x \) in the interval \( -3 < x < -2 \).
2. **Second Piece:** \( m(x) = x \) for \( -2 \le x < 2 \)
- This piece represents a line with a slope of 1 and a y-intercept of 0, passing through every point \( (x, y) \) where \( y = x \) within the interval \( -2 \le x < 2 \).
3. **Third Piece:** \( m(x) = -x^2 + 4 \) for \( x \ge 2 \)
- This piece is a downward-opening parabola with a vertex at \( (0, 4) \). For this specific interval, we start considering this function only for \( x \ge 2 \).
### Additional Graph Information:
There is a partially visible graph below the text depicting a coordinate plane labeled with the y-axis and visible ticks at intervals. The grid lines on the graph suggest it is ready for plotting the given function.
#### Plotting Steps:
1. **For \( -3 < x < -2 \):** Draw a horizontal line at \( y = -3 \).
2. **For \( -2 \le x < 2 \):** Draw a straight line passing through \( (-2, -2) \) and ending at points just before \( (2, 2) \).
3. **For \( x \ge 2 \):** Start plotting the parabolic curve \( y = -x^2 + 4 \) from \( x = 2 \) onwards.
After completing these steps, you will have a complete graph of the piecewise function \( m(x) \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ffd9792-2b35-4da7-9b44-f091980b5633%2Fb1afa74e-bc88-4414-b5a4-5bfb6c900b77%2F2bgvc5u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Graph the Function**
Consider the piecewise function \( m(x) \) defined as follows:
\[
m(x) =
\begin{cases}
-3 & \text{for } -3 < x < -2 \\
x & \text{for } -2 \le x < 2 \\
-x^2 + 4 & \text{for } x \ge 2
\end{cases}
\]
### Explanation of the Piecewise Function:
1. **First Piece:** \( m(x) = -3 \) for \( -3 < x < -2 \)
- This piece of the function is a constant, where the function value is -3 for all \( x \) in the interval \( -3 < x < -2 \).
2. **Second Piece:** \( m(x) = x \) for \( -2 \le x < 2 \)
- This piece represents a line with a slope of 1 and a y-intercept of 0, passing through every point \( (x, y) \) where \( y = x \) within the interval \( -2 \le x < 2 \).
3. **Third Piece:** \( m(x) = -x^2 + 4 \) for \( x \ge 2 \)
- This piece is a downward-opening parabola with a vertex at \( (0, 4) \). For this specific interval, we start considering this function only for \( x \ge 2 \).
### Additional Graph Information:
There is a partially visible graph below the text depicting a coordinate plane labeled with the y-axis and visible ticks at intervals. The grid lines on the graph suggest it is ready for plotting the given function.
#### Plotting Steps:
1. **For \( -3 < x < -2 \):** Draw a horizontal line at \( y = -3 \).
2. **For \( -2 \le x < 2 \):** Draw a straight line passing through \( (-2, -2) \) and ending at points just before \( (2, 2) \).
3. **For \( x \ge 2 \):** Start plotting the parabolic curve \( y = -x^2 + 4 \) from \( x = 2 \) onwards.
After completing these steps, you will have a complete graph of the piecewise function \( m(x) \
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