Question
Graph the piecewise function given below.
f, of, x, equals, left curly bracket, begin cases, begin case 1, minus, 4, for , x, is less than or equal to, 4, end case 1, begin case 2, left parenthesis, x, minus, 5, right parenthesis, squared, for , 4, is less than, x, is less than or equal to, 7, end case 2, end cases, .
f(x)={
−4
(x−5)
2
for x≤4
for 4<x≤7
f, of, x, equals, left curly bracket, begin cases, begin case 1, minus, 4, for , x, is less than or equal to, 4, end case 1, begin case 2, left parenthesis, x, minus, 5, right parenthesis, squared, for , 4, is less than, x, is less than or equal to, 7, end case 2, end cases, .
f(x)={
−4
(x−5)
2
for x≤4
for 4<x≤7
Answers
GPT-4o mini
To graph the piecewise function \( f(x) \) defined as:
\[
f(x) =
\begin{cases}
-4 & \text{for } x \leq 4 \\
(x - 5)^2 & \text{for } 4 < x \leq 7
\end{cases}
\]
### Step 1: Define the pieces of the function
1. **For \( x \leq 4 \)**:
- The function is constant at \( f(x) = -4 \). This means for all values of \( x \) less than or equal to 4, the graph will have a horizontal line at \( y = -4 \).
2. **For \( 4 < x \leq 7 \)**:
- The function is quadratic given by \( f(x) = (x - 5)^2 \). This is a parabola that opens upwards.
- Since the vertex of the parabola is at \( x = 5 \) (where \( (x-5)^2 = 0 \)), the value of the function at the vertex is \( f(5) = (5 - 5)^2 = 0 \).
### Step 2: Find key points in the quadratic piece
- For \( x = 4.5 \):
\[
f(4.5) = (4.5 - 5)^2 = (-0.5)^2 = 0.25
\]
- For \( x = 6 \):
\[
f(6) = (6 - 5)^2 = (1)^2 = 1
\]
- For \( x = 7 \):
\[
f(7) = (7 - 5)^2 = (2)^2 = 4
\]
### Step 3: Sketch the graph
1. **For the constant part**:
- Draw a horizontal line at \( y = -4 \) from the far left up to \( x = 4 \) and include the point (4, -4) as part of the graph.
2. **For the quadratic part**:
- At \( x = 4 \), the function transitions to the quadratic part:
- At \( x = 5 \), the point is (5, 0).
- At \( x = 4.5 \), the point is (4.5, 0.25).
- At \( x = 6 \), the point is (6, 1).
- At \( x = 7 \), the point is (7, 4).
- Draw the curve from (4, -4) upwards, starting near \( (4.5, 0.25) \) and reaching \( (7, 4) \).
3. **End behavior**:
- The quadratic piece continues to increase as \( x \) increases towards \( 7 \).
### Conclusion
In the final graph, you should see a horizontal line segment at \( y = -4 \) for \( x \leq 4 \), transitioning at \( (4, -4) \) to the curve of the quadratic function which completes at \( (7, 4) \) with an open circle at \( (4, -4) \) to indicate that this point is included in the first part of the piecewise function.
The result is a graph with two distinct parts that reflect the piecewise definition given.
\[
f(x) =
\begin{cases}
-4 & \text{for } x \leq 4 \\
(x - 5)^2 & \text{for } 4 < x \leq 7
\end{cases}
\]
### Step 1: Define the pieces of the function
1. **For \( x \leq 4 \)**:
- The function is constant at \( f(x) = -4 \). This means for all values of \( x \) less than or equal to 4, the graph will have a horizontal line at \( y = -4 \).
2. **For \( 4 < x \leq 7 \)**:
- The function is quadratic given by \( f(x) = (x - 5)^2 \). This is a parabola that opens upwards.
- Since the vertex of the parabola is at \( x = 5 \) (where \( (x-5)^2 = 0 \)), the value of the function at the vertex is \( f(5) = (5 - 5)^2 = 0 \).
### Step 2: Find key points in the quadratic piece
- For \( x = 4.5 \):
\[
f(4.5) = (4.5 - 5)^2 = (-0.5)^2 = 0.25
\]
- For \( x = 6 \):
\[
f(6) = (6 - 5)^2 = (1)^2 = 1
\]
- For \( x = 7 \):
\[
f(7) = (7 - 5)^2 = (2)^2 = 4
\]
### Step 3: Sketch the graph
1. **For the constant part**:
- Draw a horizontal line at \( y = -4 \) from the far left up to \( x = 4 \) and include the point (4, -4) as part of the graph.
2. **For the quadratic part**:
- At \( x = 4 \), the function transitions to the quadratic part:
- At \( x = 5 \), the point is (5, 0).
- At \( x = 4.5 \), the point is (4.5, 0.25).
- At \( x = 6 \), the point is (6, 1).
- At \( x = 7 \), the point is (7, 4).
- Draw the curve from (4, -4) upwards, starting near \( (4.5, 0.25) \) and reaching \( (7, 4) \).
3. **End behavior**:
- The quadratic piece continues to increase as \( x \) increases towards \( 7 \).
### Conclusion
In the final graph, you should see a horizontal line segment at \( y = -4 \) for \( x \leq 4 \), transitioning at \( (4, -4) \) to the curve of the quadratic function which completes at \( (7, 4) \) with an open circle at \( (4, -4) \) to indicate that this point is included in the first part of the piecewise function.
The result is a graph with two distinct parts that reflect the piecewise definition given.