Precalculus Exam 1 Preparation: Key Concepts and Problem Types

Linear Equations and Graphs

Finding Intercepts and Slope

Linear equations can be graphed by identifying their x- and y-intercepts and slope. The general form of a linear equation is Ax + By = C.

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

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

• Slope: Rearrange to slope-intercept form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Example: For $8x - 17y = 7$:

• Rearrange: $y = \frac{8}{17}x - \frac{7}{17}$

• Slope: $m = \frac{8}{17}$

• X-intercept: $x = \frac{7}{8}$

• Y-intercept: $y = -\frac{7}{17}$

Application: Graphing the line using intercepts and slope.

Geometry: Circles and Distance

Endpoints of a Diameter and Equation of a Circle

The equation of a circle with center $(h, k)$ and radius $r$ is:

• $(x - h)^2 + (y - k)^2 = r^2$

Given endpoints of a diameter, the center is the midpoint, and the radius is half the distance between endpoints.

• Midpoint: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Distance: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Radius: $r = \frac{d}{2}$

Example: Endpoints $(5,6)$ and $(-1,2)$:

• Midpoint: $(2,4)$

• Distance: $\sqrt{(5 - (-1))^2 + (6 - 2)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}$

• Radius: $\sqrt{13}$

• Equation: $(x - 2)^2 + (y - 4)^2 = 13$

Graphing Absolute Value Functions

Transformations of Absolute Value Functions

Absolute value functions can be transformed by shifting, stretching, or reflecting. The parent function is $y = |x|$.

• Vertical shift: $y = |x| + k$ shifts up/down by $k$.

• Horizontal shift: $y = |x - h|$ shifts right/left by $h$.

• Reflection: $y = -|x|$ reflects over the x-axis.

• Vertical stretch/compression: $y = a|x|$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

Example: $f(x) = 6 - 2|x - 4|$

• Parent function: $y = |x|$

• Shift right by 4, reflect over x-axis, stretch by 2, shift up by 6.

Function Operations and Composition

Evaluating and Composing Functions

Functions can be added, subtracted, multiplied, divided, or composed. Composition involves substituting one function into another.

• Addition: $(f + g)(x) = f(x) + g(x)$

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

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

Example: $f(x) = 3x - 2$, $g(x) = x^2 + x$

• $(f \circ g)(2) = f(g(2)) = f(6) = 3 \times 6 - 2 = 16$

• $(g \circ f)(2) = g(f(2)) = g(4) = 4^2 + 4 = 20$

Solving Logarithmic Equations

Properties and Solutions of Logarithmic Equations

Logarithmic equations often require combining logs and using exponent rules. The logarithm $\log_b(x)$ is the inverse of $b^x$.

• Product Rule: $\log_b(A) + \log_b(B) = \log_b(AB)$

• Exponentiate both sides to solve: If $\log_b(x) = y$, then $x = b^y$.

Example: $\log_3(7 - x) + \log_3(2x + 1) = 3$

• Combine: $\log_3[(7 - x)(2x + 1)] = 3$

• Exponentiate: $(7 - x)(2x + 1) = 3^3 = 27$

• Solve: $2x^2 + x + 14x + 7 = 27 \implies 2x^2 + 15x + 7 - 27 = 0 \implies 2x^2 + 15x - 20 = 0$

• Find roots and check for valid solutions (argument of log must be positive).

Domain of Functions

Finding the Domain

The domain of a function is the set of all input values for which the function is defined. Restrictions include division by zero and taking square roots of negative numbers.

• Rational functions: Denominator cannot be zero.

• Radical functions: Expression under even root must be non-negative.

Example: $f(x) = \frac{x^2 - 9}{x - 3}$

• Domain: $x - 3 \neq 0 \implies x \neq 3$

Example: $f(x) = \frac{5x - 11}{\sqrt{8 - x}}$

• Domain: $8 - x > 0 \implies x < 8$

Application: Always check for division by zero and negative radicands.