Precalculus Midterm Exam Review: Key Concepts and Practice Problems

Functions and Their Properties

Intercepts and Graphing Linear Equations

Understanding how to find intercepts and graph linear equations is fundamental in precalculus. Intercepts are points where the graph crosses the axes.

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

• Graphing: Plot the intercepts and draw the line through them.

• Example: For $5x - 2y = 10$, set $y = 0$ to find $x$-intercept, and $x = 0$ to find $y$-intercept.

Evaluating Functions

Functions assign each input exactly one output. Evaluating a function means substituting a given value for the variable.

• Example: Given $f(x) = 2x^2 - x + 5$, evaluate $f(-1)$:

• $f(-1) = 2(-1)^2 - (-1) + 5 = 2(1) + 1 + 5 = 8$

Finding the Equation of a Line

To find the equation of a line given two points, use the point-slope form and calculate the slope.

• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Point-slope form: $y - y_1 = m(x - x_1)$

• Example: Find the equation for points $(4, -10)$ and $(-8, 12)$.

Parallel and Perpendicular Lines

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Parallel: Same slope as the given line.

• Perpendicular: Slope is $-1/m$ if the original slope is $m$.

• Example: Find the equation of the line parallel to $2x + 3y = -6$ passing through $(-1,3)$.

Quadratic Functions and Applications

Finding Zeros of Functions

The zeros of a function are the values of $x$ for which $f(x) = 0$.

• Set the function equal to zero and solve for $x$.

• Example: $f(x) = 3x + 9$; set $3x + 9 = 0$ to find $x = -3$.

Quadratic Equations and Vertex Form

Quadratic functions can be written in standard or vertex form. The vertex form is useful for identifying the vertex and axis of symmetry.

• Standard form: $y = ax^2 + bx + c$

• Vertex form: $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

• Axis of symmetry: $x = h$

• Example: For $y = 5x^2 - 10x + 7$, complete the square to write in vertex form.

Applications: Motion and Optimization

Quadratic functions model projectile motion and optimization problems.

• Height of an object: $h(t) = -16t^2 + 24t$ models the height after $t$ seconds.

• Maximum height: Occurs at $t = -\frac{b}{2a}$ for $h(t) = at^2 + bt + c$.

• Example: Find the maximum height and when it occurs for $h(t) = -16t^2 + 24t$.

Inequalities and Interval Notation

Solving Linear and Quadratic Inequalities

Inequalities express a range of possible values. Solutions are often written in interval notation.

• Linear inequality: Solve as you would an equation, but reverse the inequality when multiplying/dividing by a negative.

• Quadratic inequality: Factor and test intervals.

• Interval notation: $(a, b)$ for $a < x < b$; $[a, b]$ for $a \leq x \leq b$.

• Example: Solve $2x - 1 \leq 8$; $x \leq 4.5$.

Function Operations and Domains

Domain and Asymptotes

The domain of a function is the set of all possible input values. Asymptotes are lines the graph approaches but never touches.

• Domain: Exclude values that make the denominator zero or result in negative radicands for even roots.

• Vertical asymptote: Set denominator equal to zero.

• Horizontal asymptote: Compare degrees of numerator and denominator.

• Example: For $f(x) = \frac{x^2 + 3x + 1}{4x^2 + 9}$, domain is all real $x$; no vertical asymptotes since denominator never zero.

Function Composition and Difference Quotient

Function composition combines two functions. The difference quotient is used to find the average rate of change.

• Composition: $(f \circ g)(x) = f(g(x))$

• Difference quotient: $\frac{m(x + h) - m(x)}{h}$

• Example: For $m(x) = x^2 - 5$, compute $\frac{m(x + h) - m(x)}{h}$.

Polynomials and Division

Polynomial Division

Dividing polynomials can be done using long division or synthetic division. The result is written as $f(x) = Q(x) + \frac{r(x)}{d(x)}$.

• Dividend: The polynomial being divided.

• Divisor: The polynomial you divide by.

• Quotient: The result of the division.

• Remainder: What is left after division.

• Example: Divide $x^2 - 2x^2 + 3$ by $x + 2$.

Radicals and Complex Numbers

Simplifying Radicals and Complex Expressions

Radicals and complex numbers often require simplification to standard form $a + bi$.

• Radical simplification: $\sqrt{27} + 3$ can be simplified to $3\sqrt{3} + 3$.

• Complex numbers: $a + bi$ form, where $i = \sqrt{-1}$.

• Example: $(3 + 4i)/(3 - 4i)$ can be simplified by multiplying numerator and denominator by the conjugate.

Applications: Rates, Geometry, and Word Problems

Rates of Change

Average rate of change measures how a quantity changes over time.

• Formula: $\text{Average Rate} = \frac{f(b) - f(a)}{b - a}$

• Example: Minimum wage increased from $3.10$ in 1980 to $5.15$ in 2006. Average rate: $\frac{5.15 - 3.10}{2006 - 1980}$

Geometry and Optimization

Problems may involve finding dimensions, maximizing area, or minimizing cost.

• Example: A box with a square base and open top: If the base is $x$ cm and the height is $h$ cm, and the volume is $V$, then $V = x^2 h$.

Mixture and Interest Problems

Mixture problems involve combining quantities; interest problems use formulas for simple and compound interest.

• Simple interest: $I = Prt$

• Compound interest: $A = P(1 + r/n)^{nt}$

• Example: $800$ deposited into two accounts with different interest rates; set up equations to solve for amounts.

Summary Table: Key Concepts

Topic	Key Formula	Example
Linear Equation	$y = mx + b$	Find slope between $(1,2)$ and $(3,4)$
Quadratic Vertex	$x = -\frac{b}{2a}$	Vertex of $y = 2x^2 - 4x + 1$
Average Rate of Change	$\frac{f(b) - f(a)}{b - a}$	Change in wage from 1980 to 2006
Difference Quotient	$\frac{f(x + h) - f(x)}{h}$	For $f(x) = x^2$
Simple Interest	$I = Prt$	$1000$at $5\%$ for $2$ years

Additional info:

• Some questions involve graphing, interpreting solutions, and expressing answers in interval notation.

• Word problems cover rates, geometry, and financial applications, which are standard in precalculus.

• Complex numbers and radical simplification are included, as well as polynomial division and asymptotes.