Questions: Problem 3: Sketch a graph of a function that satisfies the following conditions (note: there is no one unique answer), and be sure to label all the points: a. a continuous function with an absolute minimum of -3 at x=-2 and a local minimum of -1 at x=2. b. a continuous function with no absolute extrema but with a local minimum of -2 at x=-1 and a local maximum of 2 at x=1. c. a continuous function that is increasing on (-∞, 2) and decreasing on (2,+∞).

Problem 3: Sketch a graph of a function that satisfies the following conditions (note: there is no one unique answer), and be sure to label all the points:

a. a continuous function with an absolute minimum of -3 at x=-2 and a local minimum of -1 at x=2.
b. a continuous function with no absolute extrema but with a local minimum of -2 at x=-1 and a local maximum of 2 at x=1.
c. a continuous function that is increasing on (-∞, 2) and decreasing on (2,+∞).

Transcript text: Problem 3: Sketch a graph of a function that satisfies the following conditions (note: there is no one unique answer), and be sure to label all the points: Page 1 of 2 $\qquad$ Math 22 - College Algebra with Analytic Geometry and Applications II Homework 2 a. a continuous function with an absolute minimum of -3 at $x=-2$ and a local minimum of -1 at $x=2$. b. a continuous function with no absolute extrema but with a local minimum of -2 at $x=-1$ and a local maximum of 2 at $x=1$. c. a continuous function that is increasing on $(-\infty, 2)$ and decreasing on $(2,+\infty)$.

Solution

Solution Steps

Step 1: Identify the conditions for the function

We need to create a continuous function with the following properties:

An absolute minimum of -3 at $ x = -2 $ and a local minimum of -1 at $ x = 2 $.

Step 2: Construct a function that meets the conditions

One possible function that meets these conditions is: \[ y = (x + 2)^2 - 3 \]

Step 3: Identify the conditions for the second function

We need to create a continuous function with the following properties:

No absolute extrema but with a local minimum of -2 at $ x = -1 $ and a local maximum of 2 at $ x = 1 $.

Step 4: Construct a function that meets the conditions

One possible function that meets these conditions is: \[ y = -x^3 + 3x \]

Step 5: Identify the conditions for the third function

We need to create a continuous function with the following properties:

Increasing on $ (-\infty, 2) $ and decreasing on $ (2, +\infty) $.

Step 6: Construct a function that meets the conditions

One possible function that meets these conditions is: \[ y = -x^2 + 4x \]

Final Answer

{"axisType": 3, "coordSystem": {"xmin": -5, "xmax": 5, "ymin": -5, "ymax": 5}, "commands": ["y = (x + 2)^2 - 3", "y = -x^3 + 3x", "y = -x^2 + 4x"], "latex_expressions": ["$y = (x + 2)^2 - 3$", "$y = -x^3 + 3x$", "$y = -x^2 + 4x$"]}

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