Calculus I - Type of Functions

• Mathematics •

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### 1. Constant and Linear Functions * **Constant Functions ($y = c$):** Produce a single output value regardless of the input, graphing as a horizontal line at height $c$. They correspond to a polynomial of degree $0$ ($f(x) = a$ for $a \neq 0$). * **Linear Functions ($y = mx + b$):** Graph as straight lines using slope-intercept form. * **$y$-intercept ($b$):** The value of the function at $f(0)$ where the line crosses the vertical axis. * **Slope ($m$):** Measures the steepness of the line. Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope formula is: $$m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$$ --- ### 2. Polynomial Functions A general polynomial has the standard form: $$f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$ * **Domain & Range:** Defined for all real numbers. If the degree $n$ (where $a_n \neq 0$) is **odd**, the range is all real numbers. If $n$ is **even**, the function reaches a global maximum (if $a_n < 0$) or a global minimum (if $a_n > 0$). * **Intercepts, Roots & Turns:** The constant term $a_0$ is the $y$-intercept. A degree-$n$ polynomial has at most $n$ roots (zeroes where the graph crosses the $x$-axis) and at most $n - 1$ "turns" (local peaks or valleys). * **Quadratic Functions (Degree 2):** Written as $f(x) = ax^2 + bx + c$, graphing as a **parabola**. * **Vertex Form:** $f(x) = a(x - h)^2 + k$, where the vertex is at $(h, k)$ with $h = -\frac{b}{2a}$ and $k = f(h)$. If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward. A larger $\vert{}a\vert{}$ makes the parabola narrower, while a smaller $\vert{}a\vert{}$ makes it wider. * **Roots via the Quadratic Formula:** Found using $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The discriminant ($b^2 - 4ac$) determines the number of real roots: $>0$ yields 2 real roots, $=0$ yields 1 real root, and $<0$ yields no real roots. --- ### 3. Rational Functions Defined as the quotient of two polynomials: $$f(x) = \frac{p(x)}{q(x)}$$ * **Domain & Zeroes:** Defined everywhere except at the roots of the denominator $q(x)$. The zeroes of $f(x)$ are the roots of $p(x)$ that are *not* also roots of $q(x)$. * **End Behavior (Large $\vert{}x\vert{}$):** * If $\deg p(x) \ge \deg q(x)$, at very large $x$ the function behaves like a polynomial of degree $\deg p(x) - \deg q(x)$. * If $\deg p(x) < \deg q(x)$, $f(x)$ approaches $0$ as $\vert{}x\vert{}$ becomes very large (creating a horizontal asymptote at $y = 0$). --- ### 4. Exponential and Logarithmic Functions These functions model very fast or very slow growth and are inverses of one another. * **Exponential Functions ($y = a^x$, where $a > 0, a \neq 1$):** * **Domain & Range:** The domain is all real numbers; the range is positive real numbers $(0, \infty)$. * **Behavior:** Always increasing if $a > 1$ and always decreasing if $a < 1$. They grow faster than any polynomial. * **Logarithmic Functions ($y = \log_a x$):** * **Definition:** $\log_a x = y$ if and only if $a^y = x$ ("the power to which you raise $a$ to get $x$"). * **Domain:** Defined only for positive real numbers. * **Natural Logarithm:** Uses the special base $e \approx 2.71828$, written as $\log_e x = \ln x$. * **Change of Base:** $\log_a x = (\log_a b)(\log_b x) = \frac{\ln x}{\ln a}$, and $\log_a b = \frac{1}{\log_b a}$. | Exponent Rules| Logarithm Rules| | --- | --- | | $a^{m+n} = a^m a^n$<br> | $\log_a(bc) = \log_a b + \log_a c$<br> | | $a^{m-n} = \frac{a^m}{a^n}$ and $a^{-n} = \frac{1}{a^n}$<br> | $\log_a\left(\frac{b}{c}\right) = \log_a b - \log_a c$ and $\log_a\left(\frac{1}{b}\right) = -\log_a b$<br> | | $(a^m)^n = a^{mn}$<br> | $\log_a(b^n) = n \log_a b$<br> | | $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$<br> | $\log_a\left(\sqrt[n]{b^m}\right) = \frac{m}{n} \log_a b$<br> | | $(ab)^n = a^n b^n$ and $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$<br> | $\log_a(a^n) = n$ and $a^{\log_a b} = b$<br> | --- ### 5. Trigonometric Functions Defined via a point $P = (x, y)$ on the **unit circle** (radius 1 centered at the origin) at angle $\theta$ measured counterclockwise from $A = (1, 0)$. Angles convert via $\frac{\theta \text{ (degrees)}}{180} = \frac{\theta \text{ (radians)}}{\pi}$. * **Six Core Ratios:** * $\sin \theta = y$ (odd function, range $[-1, 1]$) * $\cos \theta = x$ (even function, range $[-1, 1]$) * $\tan \theta = \frac{y}{x} = \frac{\sin \theta}{\cos \theta}$ (slope of line $OP$) * $\sec \theta = \frac{1}{\cos \theta} = \frac{1}{x}$, $\csc \theta = \frac{1}{\sin \theta} = \frac{1}{y}$, $\cot \theta = \frac{1}{\tan \theta} = \frac{x}{y}$ * **General Sinusoidal Wave Form:** $$y = A \sin B(x - h) + k$$ * **$A$ (Amplitude):** Half the distance between maximum and minimum values. * **$k$ (Average Value):** Vertical center halfway between maximum and minimum. * **$\frac{2\pi}{B}$ (Period):** Length of one full cycle (a larger $B$ means more cycles in a given interval). * **$h$ (Phase Shift):** Horizontal shift showing how far the start of the cycle is from the $y$-axis. --- ### 6. Combining, Transforming, and Inverting Functions * **Arithmetic & Composition:** Functions can be added, subtracted, multiplied, or divided on the intersection of their domains (excluding denominator zeroes for division). They can also be composed as $(f \circ g)(x) = f(g(x))$ where $g(x)$ is applied first. * **Geometric Transformations of $f(x)$:** 1. **Translations:** $y = f(x) + c$ shifts vertically ($+$ up, $-$ down); $y = f(x - c)$ shifts horizontally ($c > 0$ right, $c < 0$ left). 2. **Scaling:** $y = c f(x)$ stretches vertically if $c > 1$ (compresses if $c < 1$); $y = f\left(\frac{x}{c}\right)$ stretches horizontally if $c > 1$ (compresses if $c < 1$). 3. **Reflections & Symmetry:** $y = -f(x)$ reflects across the $x$-axis; $y = f(-x)$ reflects across the $y$-axis. (Note: The SparkChart contains a known typo on page 2 swapping the axis names in the prose, though its accompanying diagrams show the reflections accurately.) **Even functions** satisfy $f(x) = f(-x)$ (symmetric across the $y$-axis, like $\cos x$), while **odd functions** satisfy $f(x) = -f(-x)$ ($180^\circ$ rotational symmetry around the origin, like $\sin x$). * **Inverse Functions ($f^{-1}(x)$):** Exist if $f(x)$ never takes the same value twice (passes the **horizontal line test**), satisfying $f^{-1}(f(x)) = x$ and graphing as a reflection across the line $y = x$. * **Inverse Trigonometric Functions:** Created by restricting periodic trig functions to a single one-to-one cycle: * **Arcsine ($\sin^{-1} x$):** Domain $[-1, 1]$, Range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. * **Arccosine ($\cos^{-1} x$):** Domain $[-1, 1]$, Range $[0, \pi]$. * **Arctangent ($\tan^{-1} x$):** Domain $(-\infty, \infty)$, Range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.