Precalculus Exam 1 Review: Quadratic, Polynomial, and Exponential Functions

Quadratic Functions

Understanding Quadratic Functions

Quadratic functions are polynomial functions of degree 2, typically written in the form $f(x) = ax^2 + bx + c$. They model many real-world phenomena, such as the trajectory of a ball.

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

• Vertex Form: $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola

• Factored Form: $f(x) = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots

Key Properties

• Vertex: The highest or lowest point of the parabola, found at $x = -\frac{b}{2a}$

• Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$

• Direction: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward

• Y-intercept: The point $(0, c)$

• X-intercepts (Roots): Solutions to $ax^2 + bx + c = 0$

Example: Ball Trajectory

The height $h$ (in meters) of a ball as a function of horizontal distance $d$ (in meters) is given by:

$h(d) = -0.02d^2 + 0.74d + 1.3$

• Initial Height: $h(0) = 1.3$ meters

• Maximum Height: Occurs at $d = -\frac{0.74}{2 \times -0.02} = 18.5$ meters

• Maximum Value: Substitute $d = 18.5$ into $h(d)$ to find the maximum height

• Domain: Values of $d$ for which $h(d) \geq 0$ (the ball is above the ground)

• Range: $[0, \text{maximum height}]$

Concavity

• If $a > 0$, the function is concave up (U-shaped)

• If $a < 0$, the function is concave down (n-shaped)

Graphing Quadratic Functions

• Plot the vertex, axis of symmetry, and intercepts

• Sketch the parabola based on the direction of opening

Polynomial Functions

General Properties

Polynomial functions are sums of terms of the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0$, where $n$ is a non-negative integer.

• Degree: The highest power of $x$

• Leading Coefficient: The coefficient of the highest degree term

• End Behavior: Determined by the degree and leading coefficient

• Zeros: Values of $x$ where the function equals zero

Example: Cubic Polynomial

$p(x) = (x + 1)(x - 5)(x - 2)$

• Degree: 3 (cubic)

• Zeros: $x = -1, 2, 5$

• End Behavior: As $x \to \infty$, $p(x) \to \infty$; as $x \to -\infty$, $p(x) \to -\infty$

Graphing Polynomials

• Plot zeros and y-intercept

• Determine end behavior from degree and leading coefficient

• Sketch the curve, noting turning points (maximum of $n-1$ for degree $n$)

Exponential Functions

Definition and Properties

An exponential function has the form $f(x) = ab^{x}$, where $a \neq 0$ and $b > 0$, $b \neq 1$.

• Growth: If $b > 1$, the function grows as $x$ increases

• Decay: If $0 < b < 1$, the function decays as $x$ increases

• Y-intercept: $(0, a)$

• Horizontal Asymptote: $y = 0$

Growth Factor and Rate

• Growth Factor: $b = 1 + r$, where $r$ is the growth rate (as a decimal)

• Percent Rate of Change: $r = b - 1$

Example:

• For a 5% growth rate: $b = 1.05$

• For a 10% decay rate: $b = 0.90$

Matching Equations to Graphs

• Positive $a$ and $b > 1$: Increasing exponential

• Negative $a$: Reflection over the x-axis

• $0 < b < 1$: Decreasing exponential

Table Example

x	-2	-1	0	1	2
f(x)	0.16	0.4	1	2.5	6.25

Additional info: This table shows values of an exponential function for various x-values.

Applications

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

• Continuous compounding: $A = Pe^{rt}$

Logarithmic Functions

Definition and Properties

A logarithmic function is the inverse of an exponential function. It is written as $y = \log_b(x)$, which means $b^y = x$.

• Common Logarithm: Base 10, $\log(x)$

• Natural Logarithm: Base $e$, $\ln(x)$

Properties of Logarithms

• $\log_b(MN) = \log_b(M) + \log_b(N)$

• $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$

• $\log_b(M^k) = k\log_b(M)$

Solving Logarithmic and Exponential Equations

• Rewrite exponentials as logarithms and vice versa

• Example: $y = \log_b(x) \iff x = b^y$

• Example: $y = \ln(27) \implies e^y = 27$

Applications

• Half-life: $N(t) = N_0 e^{-kt}$

• Solving for time in exponential growth/decay

Order of Magnitude and the Number $e$

Understanding $e$

The number $e \approx 2.718$ is the base of the natural logarithm and arises in continuous growth and decay processes.

• Order numbers: $e^{1/3}, 2^2, 6^1, 10^1$ from smallest to largest

• Continuous growth formula: $A = Pe^{rt}$

Compound Interest and Effective Annual Rate

Formulas

• Compound Interest: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

• Continuous Compounding: $A = Pe^{rt}$

• Effective Annual Rate (EAR): $EAR = \left(1 + \frac{r}{n}\right)^n - 1$

Example:

• Find the value of a $70 investment at 4.2% continuous rate after 4 years:$A = 70e^{0.042 \times 4}$

Summary Table: Exponential and Logarithmic Equations

Equation	Equivalent Form
$y = \log_b(x)$	$x = b^y$
$y = \ln(x)$	$x = e^y$
$\log_b(MN)$	$\log_b(M) + \log_b(N)$
$\log_b(\frac{M}{N})$	$\log_b(M) - \log_b(N)$
$\log_b(M^k)$	$k\log_b(M)$

Additional info: Table summarizes key logarithmic properties and conversions.