Precalculus Midterm Exam Review: Key Concepts and Practice Problems

Functions and Graphs

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 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 given line.

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

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

Quadratic and Polynomial Functions

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 $x = -3$.

Vertex Form of a Parabola

The vertex form of a quadratic function is useful for identifying the vertex and axis of symmetry.

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

• Standard form: $y = ax^2 + bx + c$

• Axis of symmetry: $x = h$

• Example: Convert $y = 5x^2 - 10x + 7$ to vertex form.

Solving Quadratic Equations

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: Solve $5x^2 - 16 = 0$.

Inequalities and Interval Notation

Solving and Graphing Inequalities

Inequalities are solved similarly to equations, but solutions are expressed as intervals.

• Example: Solve $2x - 5 < x + 7$:

• $2x - 5 < x + 7 \implies x < 12$

• Interval notation: $(-\infty, 12)$

Rational, Radical, and Complex Expressions

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.

• Vertical asymptotes: Set denominator equal to zero.

• Horizontal asymptotes: Compare degrees of numerator and denominator.

• Example: $f(x) = \frac{x^2 - 2x - 2}{x - 2}$

Simplifying Complex Numbers

Complex numbers are expressed in the form $a + bi$, where $i = \sqrt{-1}$.

• Standard form: $a + bi$

• Example: $(3 + 4i)/(3 - 4i)$

Solving Radical Equations

Radical equations involve roots. Isolate the radical and square both sides to solve.

• Example: $\sqrt{x - 7} + 3 = x$

Applications and Word Problems

Rate of Change

Average rate of change measures how a quantity changes over an interval.

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

• Example: Minimum wage increased from $3.10$ in $1980$to $5.15$ in $2006$.

Motion and Mixture Problems

These problems use algebraic equations to model real-world scenarios.

• Distance formula: $d = rt$

• Example: A boat moves at $11$mph in still water; river current is $5$ mph.

Geometry and Area

Problems may involve finding dimensions, area, or perimeter.

• Area of rectangle: $A = lw$

• Example: Box with square piece of sheet metal, sides $7$cm, open top, volume $112$ cm$^3$.

Polynomial Division and Remainder

Polynomial Long Division

Dividing polynomials involves expressing $f(x)$ as $Q(x) + \frac{r(x)}{d(x)}$.

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

Difference Quotient

Definition and Application

The difference quotient is used to find the average rate of change and is foundational for calculus.

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

• Example: Given $m(x) = x^2 - 5$, find and simplify the difference quotient.

Summary Table: Key Concepts

Topic	Key Formula	Example
Linear Equation	$y = mx + b$	Find slope between $(1,2)$ and $(3,4)$
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	Solve $x^2 - 4x + 3 = 0$
Difference Quotient	$\frac{f(x + h) - f(x)}{h}$	Given $f(x) = x^2$, compute for $h$
Average Rate of Change	$\frac{f(b) - f(a)}{b - a}$	Minimum wage from 1980 to 2006
Distance	$d = rt$	Boat in river problem

Additional info:

• Some questions involve graphing, interpreting solutions in context, and applying algebraic methods to real-world scenarios.

• Vertex form and difference quotient are foundational for calculus and further study in mathematics.