Precalculus Study Guide: Linear, Quadratic, Exponential, and Logarithmic Functions

Linear, Quadratic, Exponential, and Logarithmic Functions

Linear Functions

Linear functions are algebraic expressions of the form f(x) = mx + b, where m is the slope and b is the y-intercept. They model relationships with constant rates of change.

• Definition: A function whose graph is a straight line.

• General Form: $f(x) = mx + b$

• Example: If $f(0) = 5$ and 3 is added to y for each $riangle x = 1$, then $m = 3$, $b = 5$, so $f(x) = 3x + 5$.

• Application: Used to model constant speed, cost, or other linear relationships.

Exponential Functions

Exponential functions have the form f(x) = ab^{x}, where a is the initial value and b is the base or growth factor. They model growth or decay processes.

• Definition: A function in which the variable appears in the exponent.

• General Form: $f(x) = ab^{x}$

• Example: If $f(0) = 5$ and y is multiplied by 3 for each $riangle x = 1$, then $a = 5$, $b = 3$, so $f(x) = 5 imes 3^{x}$.

• Application: Population growth, radioactive decay, compound interest.

Quadratic Functions

Quadratic functions are polynomials of degree 2, typically written as f(x) = ax^2 + bx + c. Their graphs are parabolas.

• Definition: A function where the highest power of x is 2.

• General Form: $f(x) = ax^2 + bx + c$

• Example: If zeros are at $x = -2$ and $x = 3$, and $f(0) = 6$, use factored form: $f(x) = a(x + 2)(x - 3)$. Plug $x = 0$ to solve for $a$:

$f(0) = a(0 + 2)(0 - 3) = a(2)(-3) = -6a = 6 \Rightarrow a = -1$

So, $f(x) = -1(x + 2)(x - 3)$

• Application: Projectile motion, area problems.

Secant Line

The secant line to a curve is a straight line that passes through two points on the curve. For $y = \log x$ on the interval [1, 100]:

• Definition: A line connecting two points $(x_1, f(x_1))$ and $(x_2, f(x_2))$.

• Formula for slope: $m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• Example: For $x_1 = 1$, $x_2 = 100$, $f(1) = \log 1 = 0$, $f(100) = \log 100 = 2$ (common log):

Slope: $m = \frac{2 - 0}{100 - 1} = \frac{2}{99}$

Equation: $y - 0 = \frac{2}{99}(x - 1)$

Population Growth Model

Exponential growth models describe populations that grow at a constant percentage rate per time period.

• General Form: $P(t) = P_0 e^{rt}$

• Where: $P_0$ is the initial population, $r$ is the growth rate (as a decimal), $t$ is time.

• Example: For 5% growth, $r = 0.05$:

$P(t) = P_0 e^{0.05t}$

• Doubling Time: Set $P(t) = 2P_0$ and solve for $t$:

$2P_0 = P_0 e^{0.05t} \Rightarrow 2 = e^{0.05t} \Rightarrow \ln 2 = 0.05t \Rightarrow t = \frac{\ln 2}{0.05}$

Graph Analysis and Properties

Linear Graphs

• Example: $y = \frac{5}{2} - 2x$

• Slope: $-2$ (decreasing function)

• Y-intercept: $\frac{5}{2}$

Exponential Graphs

• General Form: $y = a b^{x}$, $a > 0$, $0 < b < 1$

• Increasing/Decreasing: If $b > 1$, increasing; if $0 < b < 1$, decreasing.

• Concavity: Exponential functions are always concave up.

• End Behavior: As $x \to -\infty$, $y \to \infty$ (if $b > 1$); as $x \to \infty$, $y \to 0$.

Logarithmic Graphs

• General Form: $y = \log_b x$, $b > 1$

• Increasing/Decreasing: Increasing for $x > 0$

• Concavity: Concave down

• Key Points: $(1, 0)$ and $(b, 1)$

Quadratic Graphs

• General Form: $y = ax^2 + bx + c$

• Concavity: Up if $a > 0$, down if $a < 0$

• Vertex: $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$

• Example: $y = -6x^2 - 12x + 5$ is concave down, vertex at $x = -\frac{-12}{2 \times -6} = -\frac{-12}{-12} = 1$

Logarithms: Properties and Calculations

Logarithm Basics

• Definition: $\log_b a$ is the exponent to which $b$ must be raised to get $a$.

• Common Logarithm: Base 10, $\log a$

• Natural Logarithm: Base $e$, $\ln a$

Evaluating Logarithms

• $\log_4 16 = 2$ (since $4^2 = 16$)

• $\log_4 4 = 1$ (since $4^1 = 4$)

• $\log_4 (-4)$ is undefined (logarithm of a negative number is not real)

• $\log_4 \left(\frac{1}{16}\right) = -2$ (since $4^{-2} = \frac{1}{16}$)

• $\log_2 64 = 6$ (since $2^6 = 64$)

Logarithm Properties

• Product Rule: $\log_b (xy) = \log_b x + \log_b y$

• Quotient Rule: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$

• Power Rule: $\log_b (x^r) = r \log_b x$

• Change of Base: $\log_b a = \frac{\log_k a}{\log_k b}$

Simplifying Logarithmic and Exponential Expressions

• Example: $e^{\ln x + 2} = e^{\ln x} \cdot e^2 = x e^2$

• Example: $\ln(\log 10^7) = \ln(7)$ (since $\log 10^7 = 7$)

Solving Exponential and Logarithmic Equations

• Exponential Equations: $3^x = 12 \Rightarrow x = \log_3 12$

• Logarithmic Equations: $\ln(x - 1) = 3 \Rightarrow x - 1 = e^3 \Rightarrow x = e^3 + 1$

• Mixed Equations: $\log(2x + 3) = 1 + \log x \Rightarrow \log(2x + 3) - \log x = 1 \Rightarrow \log\left(\frac{2x + 3}{x}\right) = 1 \Rightarrow \frac{2x + 3}{x} = 10^1 = 10 \Rightarrow 2x + 3 = 10x \Rightarrow 8x = 3 \Rightarrow x = \frac{3}{8}$

• Natural Logarithm Equation: $\ln(30 - e^x) = 2 \Rightarrow 30 - e^x = e^2 \Rightarrow e^x = 30 - e^2 \Rightarrow x = \ln(30 - e^2)$

Summary Table: Function Types and Properties

Function Type	General Form	Key Properties	Example
Linear	$f(x) = mx + b$	Constant rate of change, straight line	$f(x) = 3x + 5$
Quadratic	$f(x) = ax^2 + bx + c$	Parabola, vertex, zeros	$f(x) = -1(x + 2)(x - 3)$
Exponential	$f(x) = ab^x$	Growth/decay, constant percent change	$f(x) = 5 \times 3^x$
Logarithmic	$f(x) = \log_b x$	Inverse of exponential, slow growth	$f(x) = \log x$

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Examples and formulas are provided for each function type to aid understanding.