Question
f(x)=2(x-1)(x+3)^2
end behavior
as x goes to negative infnity f(x) goes to negative infinity
as x goes to positive infinity f(x) goes to positive infinity
roots with mutiplicity x-3 mutiplicty of 2
x=1 mutipplicty of 1
value with leading coeffcient 2
domain negative infinity positive infinity
range all real numbers
pls tell me where to graph the coordinates pls
end behavior
as x goes to negative infnity f(x) goes to negative infinity
as x goes to positive infinity f(x) goes to positive infinity
roots with mutiplicity x-3 mutiplicty of 2
x=1 mutipplicty of 1
value with leading coeffcient 2
domain negative infinity positive infinity
range all real numbers
pls tell me where to graph the coordinates pls
Answers
GPT-4o mini
To graph the function \( f(x) = 2(x-1)(x+3)^2 \) and understand its characteristics, we can analyze some key components, including its roots, behavior, and critical points.
### Key Information
1. **End Behavior**:
- As \( x \to -\infty \), \( f(x) \to -\infty \)
- As \( x \to +\infty \), \( f(x) \to +\infty \)
2. **Roots and Their Multiplicities**:
- \( x = 1 \) (multiplicity 1) — This root indicates that the curve will cross the x-axis at this point.
- \( x = -3 \) (multiplicity 2) — Since this root has an even multiplicity, the graph touches the x-axis at this point and turns back without crossing.
3. **Leading Coefficient**:
- The leading coefficient is positive (2), indicating that the ends of the graph will rise as \( x \to +\infty \) and drop as \( x \to -\infty \).
4. **Domain**:
- The domain of \( f(x) \) is all real numbers: \( (-\infty, \infty) \)
5. **Range**:
- The range of \( f(x) \) is all real numbers: \( (-\infty, \infty) \)
### Key Points for Graphing
1. **Roots**:
- **\( (-3, 0) \)**: The graph touches the x-axis.
- **\( (1, 0) \)**: The graph crosses the x-axis.
2. **Y-Intercept**:
- To find the y-intercept, set \( x = 0 \):
\[ f(0) = 2(0-1)(0+3)^2 = 2(-1)(9) = -18 \]
- So the y-intercept is **\( (0, -18) \)**.
3. **Critical Points**:
- Calculate the derivative \( f'(x) \) to find local maxima or minima.
- Use the product rule to differentiate if necessary, but we can also inspect the graph around the roots above.
4. **Behavior Between Roots**:
- For \( x < -3 \): As \( x \) approaches \( -3 \), \( f(x) \) is negative and decreases to approach 0.
- For \( -3 < x < 1 \): The function is negative, moving from 0 (at -3) to a minimum before crossing at (1,0).
- For \( x > 1 \): The function increases and rises positively towards infinity.
### Graphing
1. Plot the key coordinates:
- Roots: \( (-3,0) \), \( (1,0) \)
- Y-Intercept: \( (0, -18) \)
2. Sketch the overall shape:
- Starts from low (-∞) as \( x \) approaches \( -\infty \).
- Rises towards \( (-3,0) \), touches and turns back down.
- Falls to a minimum point and crosses the x-axis at \( (1, 0) \).
- Finally, rises to infinity as \( x \) goes to +∞.
### Summary
Using the information above, you now have the key coordinates, behavior at infinity, critical points, and a detailed description of where to plot in order to graph the function accurately. Note that the curve will have a "bounce back" effect at \( (-3, 0) \) due to the multiplicity of 2, and will cross the x-axis at \( (1, 0) \). You can use graphing software or graphing calculators to visualize this function accurately.
### Key Information
1. **End Behavior**:
- As \( x \to -\infty \), \( f(x) \to -\infty \)
- As \( x \to +\infty \), \( f(x) \to +\infty \)
2. **Roots and Their Multiplicities**:
- \( x = 1 \) (multiplicity 1) — This root indicates that the curve will cross the x-axis at this point.
- \( x = -3 \) (multiplicity 2) — Since this root has an even multiplicity, the graph touches the x-axis at this point and turns back without crossing.
3. **Leading Coefficient**:
- The leading coefficient is positive (2), indicating that the ends of the graph will rise as \( x \to +\infty \) and drop as \( x \to -\infty \).
4. **Domain**:
- The domain of \( f(x) \) is all real numbers: \( (-\infty, \infty) \)
5. **Range**:
- The range of \( f(x) \) is all real numbers: \( (-\infty, \infty) \)
### Key Points for Graphing
1. **Roots**:
- **\( (-3, 0) \)**: The graph touches the x-axis.
- **\( (1, 0) \)**: The graph crosses the x-axis.
2. **Y-Intercept**:
- To find the y-intercept, set \( x = 0 \):
\[ f(0) = 2(0-1)(0+3)^2 = 2(-1)(9) = -18 \]
- So the y-intercept is **\( (0, -18) \)**.
3. **Critical Points**:
- Calculate the derivative \( f'(x) \) to find local maxima or minima.
- Use the product rule to differentiate if necessary, but we can also inspect the graph around the roots above.
4. **Behavior Between Roots**:
- For \( x < -3 \): As \( x \) approaches \( -3 \), \( f(x) \) is negative and decreases to approach 0.
- For \( -3 < x < 1 \): The function is negative, moving from 0 (at -3) to a minimum before crossing at (1,0).
- For \( x > 1 \): The function increases and rises positively towards infinity.
### Graphing
1. Plot the key coordinates:
- Roots: \( (-3,0) \), \( (1,0) \)
- Y-Intercept: \( (0, -18) \)
2. Sketch the overall shape:
- Starts from low (-∞) as \( x \) approaches \( -\infty \).
- Rises towards \( (-3,0) \), touches and turns back down.
- Falls to a minimum point and crosses the x-axis at \( (1, 0) \).
- Finally, rises to infinity as \( x \) goes to +∞.
### Summary
Using the information above, you now have the key coordinates, behavior at infinity, critical points, and a detailed description of where to plot in order to graph the function accurately. Note that the curve will have a "bounce back" effect at \( (-3, 0) \) due to the multiplicity of 2, and will cross the x-axis at \( (1, 0) \). You can use graphing software or graphing calculators to visualize this function accurately.
sayletyler
I said I just need the coordinates
GPT-4o mini
Apologies for the misunderstanding! Here are the key coordinates for the graph of the function \( f(x) = 2(x-1)(x+3)^2 \):
1. **Roots**:
- \( (-3, 0) \) (touches the x-axis)
- \( (1, 0) \) (crosses the x-axis)
2. **Y-Intercept**:
- \( (0, -18) \)
3. **Additional Point (Local Minimum)**:
- You can find a critical point to get a sense of the shape. For example, at \( x = -1 \):
- \( f(-1) = 2(-1 - 1)(-1 + 3)^2 = 2(-2)(2)^2 = -16 \)
- Coordinate: \( (-1, -16) \)
### Summary of Coordinates
- \( (-3, 0) \)
- \( (1, 0) \)
- \( (0, -18) \)
- \( (-1, -16) \)
Feel free to add more points in-between or around these to get a more detailed graph if needed!
1. **Roots**:
- \( (-3, 0) \) (touches the x-axis)
- \( (1, 0) \) (crosses the x-axis)
2. **Y-Intercept**:
- \( (0, -18) \)
3. **Additional Point (Local Minimum)**:
- You can find a critical point to get a sense of the shape. For example, at \( x = -1 \):
- \( f(-1) = 2(-1 - 1)(-1 + 3)^2 = 2(-2)(2)^2 = -16 \)
- Coordinate: \( (-1, -16) \)
### Summary of Coordinates
- \( (-3, 0) \)
- \( (1, 0) \)
- \( (0, -18) \)
- \( (-1, -16) \)
Feel free to add more points in-between or around these to get a more detailed graph if needed!