College Algebra Test #2 Review – Step-by-Step Guidance

Q1. Graph the function: $f(x) = \frac{2}{3}x^2 - 5x + 2$

Background

Topic: Graphing Quadratic Functions

This question tests your ability to recognize and graph a quadratic function, including identifying its vertex, axis of symmetry, and intercepts.

Key Terms and Formulas:

• Quadratic function: $f(x) = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$

• Axis of symmetry: $x = -\frac{b}{2a}$

• y-intercept: $f(0)$

• x-intercepts: Solve $f(x) = 0$

Step-by-Step Guidance

1. Identify the coefficients: $a = \frac{2}{3}$, $b = -5$, $c = 2$.

2. Find the vertex using $x = -\frac{b}{2a}$.

3. Calculate the y-intercept by evaluating $f(0)$.

4. Set $f(x) = 0$ and solve for $x$ to find the x-intercepts (if any).

Try solving on your own before revealing the answer!

Q2. Graph the function: $f(x) = \frac{5}{4}x^3 + 3$

Background

Topic: Graphing Cubic Functions

This question tests your understanding of cubic functions and their general shape, intercepts, and behavior.

Key Terms and Formulas:

• Cubic function: $f(x) = ax^3 + bx^2 + cx + d$

• y-intercept: $f(0)$

• x-intercepts: Solve $f(x) = 0$

Step-by-Step Guidance

1. Identify the coefficients: $a = \frac{5}{4}$, $b = 0$, $c = 0$, $d = 3$.

2. Find the y-intercept by evaluating $f(0)$.

3. Set $f(x) = 0$ and solve for $x$ to find the x-intercepts.

4. Analyze the end behavior based on the leading coefficient and degree.

Try solving on your own before revealing the answer!

Q15. For $f(x) = 2(x + 1)^2 - 3$, find the vertex, axis of symmetry, x-intercept(s), y-intercept, and graph.

Background

Topic: Quadratic Functions in Vertex Form

This question tests your ability to analyze a quadratic function in vertex form and extract key features for graphing.

Key Terms and Formulas:

• Vertex form: $f(x) = a(x - h)^2 + k$

• Vertex: $(h, k)$

• Axis of symmetry: $x = h$

• y-intercept: $f(0)$

• x-intercepts: Solve $f(x) = 0$

Step-by-Step Guidance

1. Identify $a$, $h$, and $k$ from the equation: $a = 2$, $h = -1$, $k = -3$.

2. State the vertex as $(h, k)$.

3. Write the axis of symmetry as $x = h$.

4. Find the y-intercept by evaluating $f(0)$.

5. Set $f(x) = 0$ and solve for $x$ to find the x-intercepts.

Try solving on your own before revealing the answer!

Q20. Given $f(x) = 2x + 4$ and $g(x) = 7 - 5x$, evaluate $f(3)$.

Background

Topic: Function Evaluation

This question tests your ability to substitute a value into a function and simplify.

Key Terms and Formulas:

• Function evaluation: Substitute the given value for $x$ in the function.

Step-by-Step Guidance

1. Write the function: $f(x) = 2x + 4$.

2. Substitute $x = 3$ into the function: $f(3) = 2 \times 3 + 4$.

3. Simplify the expression to find the result.

Try solving on your own before revealing the answer!

Q28. For $f(x) = 5x^5 - 11x^4 + 10x^3 - 8x^2 - 13x + 2$, find $f(7)$ using the Remainder Theorem and synthetic division.

Background

Topic: Remainder Theorem and Synthetic Division

This question tests your ability to use synthetic division to evaluate a polynomial at a given value.

Key Terms and Formulas:

• Remainder Theorem: The remainder when $f(x)$ is divided by $(x - a)$ is $f(a)$.

• Synthetic division: A shortcut for dividing polynomials by linear factors.

Step-by-Step Guidance

1. List the coefficients of $f(x)$: $5, -11, 10, -8, -13, 2$.

2. Set up synthetic division using $x = 7$.

3. Carry out the synthetic division process step by step, bringing down the first coefficient and multiplying/adding as you go.

4. The final value you obtain is the remainder, which equals $f(7)$.

Try solving on your own before revealing the answer!