College Algebra Study Guidance: Transformations, Optimization, and Polynomial Functions

Q1. Find the function that is finally graphed after the following transformations are applied to the graph of $y = \sqrt{x}$ in the order listed:

• Reflect about the x-axis

• Shift up 5 units

• Shift right 9 units

Background

Topic: Function Transformations

This question tests your understanding of how to apply multiple transformations to a basic function, specifically the square root function. You need to know how each transformation affects the graph and the equation.

Key Terms and Formulas:

• Reflection about the x-axis: $y = -f(x)$

• Vertical shift up $k$ units: $y = f(x) + k$

• Horizontal shift right $h$ units: $y = f(x - h)$

Step-by-Step Guidance

1. Start with the base function: $y = \sqrt{x}$.

2. Apply the reflection about the x-axis: $y = -\sqrt{x}$.

3. Apply the vertical shift up 5 units: $y = -\sqrt{x} + 5$.

4. Apply the horizontal shift right 9 units: Replace $x$ with $x - 9$ to get $y = -\sqrt{x - 9} + 5$.

Try solving on your own before revealing the answer!

Final Answer: $y = -\sqrt{x - 9} + 5$

This function reflects the original square root graph about the x-axis, shifts it up by 5 units, and then shifts it right by 9 units.

Q2. A farmer has 500 meters of fencing and wants to enclose a rectangular plot that borders on a river. If the farmer does not fence the side along the river, what is the largest area that can be enclosed?

Background

Topic: Optimization (Applications of Quadratic Functions)

This question is about maximizing the area of a rectangle given a constraint on the perimeter. It is a classic optimization problem in algebra.

Key Terms and Formulas:

• Area of rectangle: $A = \text{length} \times \text{width}$

• Perimeter constraint: $2x + y = 500$ (where $x$ is the width and $y$ is the length along the river)

Step-by-Step Guidance

1. Let $x$ be the width (the two sides perpendicular to the river), and $y$ be the length (side parallel to the river).

2. Write the perimeter equation: $2x + y = 500$.

3. Solve for $y$ in terms of $x$: $y = 500 - 2x$.

4. Write the area function: $A(x) = x \cdot y = x \cdot (500 - 2x)$.

5. Simplify the area function: $A(x) = 500x - 2x^2$.

Try solving on your own before revealing the answer!

Final Answer: The largest area is $A = 500x - 2x^2$ when $x = 125$ meters, so the maximum area is $31,250$ square meters.

We found the value of $x$ that maximizes the area by completing the square or using the vertex formula for a quadratic function.

Q3. Write a polynomial function whose graph is shown (use the smallest degree possible). The coordinates of the indicated point are $(-1, -32)$.

Background

Topic: Polynomial Functions and Graphs

This question asks you to construct a polynomial function based on its graph, using the smallest degree possible and a given point.

Key Terms and Formulas:

• Degree of a polynomial: The highest power of $x$ in the function.

• Zeros of the function: Where the graph crosses the x-axis.

• General form: $f(x) = a(x - r_1)(x - r_2)...(x - r_n)$, where $r_i$ are the roots.

Step-by-Step Guidance

1. Identify the x-intercepts (roots) from the graph. These are the values where the graph crosses the x-axis.

2. Determine the degree of the polynomial by counting the number of turning points and roots.

3. Write the general form of the polynomial using the roots: $f(x) = a(x - r_1)(x - r_2)...(x - r_n)$.

4. Use the given point $(-1, -32)$ to solve for the leading coefficient $a$.

Try solving on your own before revealing the answer!

Final Answer: $f(x) = 2(x + 3)(x + 1)(x - 2)(x - 4)$

This polynomial has the correct roots and passes through the indicated point. The leading coefficient was determined using the given point.