f(x)=2(x-1)(x+3)^2

Question

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

Answer 1

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.

Answer 2

I said I just need the coordinates

Answer 3

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!