College Algebra Exam II Review – Step-by-Step Guidance

Q1. Consider $y = f(x)$ shown below.

Background

Topic: Graph Analysis of Functions

This question tests your ability to interpret a graph, identify key features such as relative minima, domain, range, and zeros of a function.

Key Terms:

• Relative minimum: A point where the function value is lower than all nearby points.

• Domain: All possible input values ($x$) for which the function is defined.

• Range: All possible output values ($y$) the function can take.

• Zero of a function: An $x$-value where $f(x) = 0$ (the graph crosses the $x$-axis).

Step-by-Step Guidance

1. To find all relative minima, look for points on the graph where the curve changes from decreasing to increasing. These are the lowest points in their immediate area.

2. To determine the domain, observe the leftmost and rightmost $x$-values for which the graph exists. Write your answer in interval notation.

3. To determine the range, find the lowest and highest $y$-values the graph reaches. Again, use interval notation.

4. To find the value(s) of $x$ where $f(x) = 0$, identify all $x$-intercepts (where the graph crosses the $x$-axis).

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Q2. Consider $f(x) = x^2 + 6x - 2$.

Background

Topic: Quadratic Functions

This question tests your ability to find the vertex of a quadratic function algebraically and to solve for $x$-intercepts.

Key Terms and Formulas:

• Vertex of a parabola: For $f(x) = ax^2 + bx + c$, the vertex is at $x = -\frac{b}{2a}$.

• $x$-intercepts: Solve $f(x) = 0$ for $x$.

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ in the quadratic: $a = 1$, $b = 6$, $c = -2$.

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

3. Plug this $x$ value back into $f(x)$ to find the $y$-coordinate of the vertex.

4. To find $x$-intercepts, set $f(x) = 0$ and solve $x^2 + 6x - 2 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

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Q3. Consider the piecewise function $f(x) = \begin{cases} -2x-4 & \text{if } -4 \leq x < 0 \\ 3x & \text{if } x \geq 0 \end{cases}$.

Background

Topic: Piecewise Functions and Graphing

This question tests your ability to graph a piecewise function and identify endpoints and other points on the graph.

Key Terms:

• Piecewise function: A function defined by different expressions over different intervals.

• Closed endpoint: The endpoint is included in the interval (solid dot).

• Open endpoint: The endpoint is not included (open circle).

Step-by-Step Guidance

1. For $-4 \leq x < 0$, use $f(x) = -2x - 4$. Calculate $f(-4)$ and $f(0)$ to find endpoints.

2. For $x \geq 0$, use $f(x) = 3x$. Calculate $f(0)$ and another value, such as $f(2)$, to plot points.

3. Identify which endpoints are closed (included) and which are open (not included) based on the interval notation.

4. Plot at least two other points from each piece to help sketch the graph accurately.

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Q4. Given $f(x) = |x|$.

Background

Topic: Transformations of Functions

This question tests your understanding of how to write equations for transformed functions, including reflections, stretches, and shifts.

Key Terms and Formulas:

• Reflection across the $x$-axis: $-f(x)$

• Horizontal stretch by $k$: $f(\frac{x}{k})$

• Vertical stretch by $a$: $a f(x)$

• Horizontal shift right by $h$: $f(x-h)$

• Vertical shift up by $k$: $f(x) + k$

Step-by-Step Guidance

1. For part (a): Reflect $f(x)$ across the $x$-axis and stretch horizontally by a factor of 2. Write the new equation using the transformation rules.

2. For part (b): Shift $f(x)$ right by 2, stretch vertically by 5, and shift up by 3. Apply each transformation step-by-step to write the new equation.

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Q5. Find a degree 3 polynomial $f(x)$ that crosses the $x$-axis at $(2,0)$ and touches the $x$-axis at $(-3,0)$.

Background

Topic: Polynomial Functions and Zeros

This question tests your ability to construct a polynomial given information about its zeros and their behavior (cross/touch).

Key Terms:

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

• Zero (root): Value of $x$ where $f(x) = 0$.

• Crosses the $x$-axis: Zero with odd multiplicity (usually 1).

• Touches the $x$-axis: Zero with even multiplicity (usually 2).

Step-by-Step Guidance

1. Write the general form of a degree 3 polynomial with zeros at $x = 2$ (crosses) and $x = -3$ (touches).

2. Assign the correct multiplicity to each zero: 1 for crossing, 2 for touching.

3. Write $f(x)$ as a product of factors based on the zeros and their multiplicities.

4. Include a leading coefficient if needed, but you can leave it as 1 unless otherwise specified.

Try solving on your own before revealing the answer!