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

Quadratic Functions

Standard, Vertex, and Factored Forms

Quadratic functions are polynomials of degree 2 and can be written in several forms, each useful for different purposes:

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

• Vertex Form: $f(x) = a(x-h)^2 + k$

• Factored Form: $f(x) = a(x - r_1)(x - r_2)$

The vertex of the parabola is at $(h, k)$ in vertex form. The axis of symmetry is $x = h$. The direction of opening is determined by the sign of $a$ (upward if $a > 0$, downward if $a < 0$).

• Maximum/Minimum: The vertex gives the maximum (if $a < 0$) or minimum (if $a > 0$) value.

• Intercepts: The y-intercept is $c$ in standard form. The x-intercepts are $r_1$ and $r_2$ in factored form.

Example: For $h(x) = -0.2x^2 + 4x + 8$, the vertex can be found by completing the square or using $x = -\frac{b}{2a}$.

Graphing Quadratic Functions

• Identify the vertex, axis of symmetry, and intercepts.

• Determine if the parabola opens upward or downward.

• Plot several points to sketch the curve.

Application: Quadratic functions model projectile motion, such as the height of a ball as a function of time or distance.

Polynomial Functions

General Properties

Polynomial functions are sums of terms of the form $a_nx^n$ where $n$ is a non-negative integer.

• Degree: The highest power of $x$.

• Leading Coefficient: The coefficient of the highest power term.

• End Behavior: Determined by the degree and leading coefficient.

Example: $p(x) = 10(x-1)^2(x-5)^3(x-2)$ is a degree 6 polynomial.

Graphing Polynomial Functions

• Identify zeros and their multiplicities.

• Analyze end behavior using the leading term.

• Sketch the graph, noting where the function crosses or touches the x-axis.

Exponential Functions

Definition and Growth Factor

An exponential function has the form $f(x) = ab^x$, where $a$ is the initial value and $b$ is the growth (or decay) factor.

• Growth Factor: For a rate of change $r$, $b = 1 + r$ (for growth), $b = 1 - r$ (for decay).

• Percent Rate of Change: $r = b - 1$

Example: For a 5% growth rate, $b = 1.05$.

Graphing Exponential Functions

• Exponential growth: $b > 1$

• Exponential decay: $0 < b < 1$

• Horizontal asymptote at $y = 0$

Equation	Graph Shape
$y = -0.25(4)^x$	Decreasing, reflected over x-axis
$y = 4(0.25)^x$	Decreasing, approaches zero
$y = -4(0.25)^x$	Decreasing, reflected over x-axis
$y = 25(4)^x$	Increasing, rapid growth

Applications

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

• Continuous growth: $A = Pe^{rt}$

• Radioactive decay: $N(t) = N_0e^{-kt}$

Example: An investment of $70$at 4.2% annual rate for 4 years:$A = 70e^{0.042 \times 4}$

Logarithmic Functions

Definition and Properties

A logarithmic function is the inverse of an exponential function: $y = \log_b x$ means $b^y = x$.

• Common Logarithm: $\log_{10} x$

• Natural Logarithm: $\ln x = \log_e x$

• Change of Base Formula: $\log_b x = \frac{\log_a x}{\log_a b}$

Solving Logarithmic Equations

• Rewrite exponential equations in logarithmic form and vice versa.

• Use properties of logarithms to simplify expressions.

Example: $y = \log_{10} x$ implies $x = 10^y$

Applications

• Half-life calculations: $t_{1/2} = \frac{\ln 2}{k}$

• Compound interest and growth problems

Additional Info

• Ordering numbers with exponents: Compare $e^{1/3}$, $2^2$, $6^1$, $10^1$ by evaluating each.

• The number $e$ is the base of natural logarithms, approximately $2.718$, and arises in continuous growth and decay models.

• Effective annual yield accounts for compounding frequency: $\text{Effective Rate} = (1 + \frac{r}{n})^n - 1$