Graph interpretation & function features
To read a value like \(g(a)\), go to \(x=a\) on the axis and find the point on the graph; the y-coordinate there is \(g(a)\). The domain is every \(x\) where the graph exists; the range is every \(y\) the graph hits (write both in interval notation). The function is increasing on intervals where the graph goes up left to right and decreasing where it goes down.
Example: Read values, domain, and range from a graph
The graph of \(h\) is shown. Compute \(h(-4),\,h(-2),\,h(0),\,h(2),\,h(4)\). State the domain and range.
Solution
\(h(-4)=1,\ h(-2)=3,\ h(0)=2,\ h(2)=-1,\ h(4)=2.\)
Domain: \([-4,4]\). Range: \([-1,3]\).
Example: Increasing/decreasing; local extrema
Using the same graph for \(h\), identify intervals where \(h\) is increasing/decreasing, and the \(x\)-values of local maxima/minima.
Solution
- Increasing on \((-4,-2)\cup(2,4)\).
- Decreasing on \((-2,2)\).
- Local max at \(x=-2\) with value \(3\).
- Local min at \(x=2\) with value \(-1\).
The difference quotient
The expression \[\dfrac{f(a+h)-f(a)}{h}\] (with \(h\neq 0\)) measures the average rate of change of \(f\) near \(x=a\). It is the slope of the secant line from \(x=a\) to \(x=a+h\). To compute it, plug in \(a\) and \(a+h\), combine into a single fraction, simplify the numerator, factor out \(h\), then cancel \(h\).
Example: Compute and simplify
Let \(f(x)=\dfrac{8}{x-5}\). Find \(f(a),\,f(a+h)\), and \(\dfrac{f(a+h)-f(a)}{h}\) (assume \(h\neq 0\)).
Solution
\(f(a)=\dfrac{8}{a-5},\quad f(a+h)=\dfrac{8}{a+h-5}\).
\begin{align*} \frac{f(a+h)-f(a)}{h} &=\frac{\frac{8}{a+h-5}-\frac{8}{a-5}}{h}\\ &=\frac{8\big[(a-5)-(a+h-5)\big]}{h(a+h-5)(a-5)}\\ &=-\frac{8}{(a+h-5)(a-5)} \end{align*}
Linear functions & equations of lines
Example: Line through a point, parallel to a given line
Find the equation of the line through \((2,-5)\) parallel to \(3x-y=7\).
Solution
Given line slope \(m=3\). Through \((2,-5)\): \(y+5=3(x-2)\Rightarrow y=3x-11\).
Example: Line through two points
Find the equation of the line through \((4,1)\) and \((-2,7)\).
Solution
Slope \(m=\dfrac{7-1}{-2-4}=\dfrac{6}{-6}=-1\). Using \((4,1)\): \(y-1=-1(x-4)\Rightarrow y=-x+5\).
Polynomial structure & basic operations
Example: Structure of a polynomial
For \(P(x)=5x^{5}-3x^{2}+7\):
- Write in standard form.
- Write the degree.
- List the coefficients (including zeros).
- Write the leading coefficient.
- Write the terms.
Solution
- (a) Already in standard form.
- (b) Degree \(5\).
- (c) Coefficients (from \(x^5\) to constant): \(5,\,0,\,0,\,-3,\,0,\,7\).
- (d) Leading coefficient \(5\).
- (e) Terms: \(5x^5,\ -3x^2,\ 7\).
Example: Simplify a polynomial expression
Simplify \((x-7)+x(x+2)-\tfrac{1}{2}x^{2}\).
Solution
\(x-7+x^2+2x-\tfrac12 x^2=\tfrac12 x^2+3x-7\).
Factoring & solving polynomial equations
The goal is to turn an equation into \(P(x)=0\) and factor so you can use the zero-product rule: if \(A\cdot B=0\), then \(A=0\) or \(B=0\).
Quadratic factoring (\(ax^2+bx+c\)): find two numbers that multiply to \(ac\) and add to \(b\) (or use the quadratic formula if needed).
Degree-four: set \(y=x^2\), factor the quadratic in \(y\), then back-substitute to get \(x\).
Grouping: split into two pairs, factor each, and look for a common binomial factor to pull out.
Rational zeros & factorization: list candidates \(\pm\) (factors of constant)/(factors of leading coeff.), test them (synthetic division), and keep factoring until you reach linear or quadratic pieces; solve remaining quadratics exactly. State all real solutions from the factored form.
Example: Quadratic factoring (solve)
Solve \(6x^{2}-x-12=0\) by factoring.
Solution
\(6x^2-x-12=(3x+4)(2x-3)=0\Rightarrow x=-\tfrac{4}{3},\ \tfrac{3}{2}\).
Example: Degree four (solve all real solutions)
Solve \(x^{4}-13x^{2}+36=0\).
Solution
Let \(y=x^2\). Then \(y^2-13y+36=(y-9)(y-4)=0\Rightarrow y=9\) or \(4\).
\(x=\pm 3,\ \pm 2\).
Example: Factor by grouping
Factor \(4x^{3}-x^{2}-16x+4\) completely.
Solution
\(x^{2}(4x-1)-4(4x-1)=(4x-1)(x^{2}-4)=(4x-1)(x-2)(x+2)\).
Example: Rational zeros & factorization
Find all rational zeros of \(Q(x)=x^{3}-4x^{2}-x+4\) and factor completely.
Solution
Rational Root Theorem: candidates \(\pm 1,\pm 2,\pm 4\). \(Q(1)=1-4-1+4=0\Rightarrow (x-1)\) is a factor.
Divide to get \(x^{2}-3x-4=(x-4)(x+1)\).
\(\boxed{Q(x)=(x-1)(x-4)(x+1)}\). Rational zeros: \(1,4,-1\).
Polynomial end behavior
Leading term controls end behavior. Odd degree: opposite ends; even degree: same ends. Sign of leading coefficient decides up/down on the right.
Example: End behavior from leading term
Examine far-left/right behavior of \(R(x)=-2x^{5}+7x^{3}-x\).
Solution
- Dominant term \(-2x^{5}\) (odd degree, negative coefficient).
- As \(x\to+\infty\), \(R(x)\to -\infty\) (right end down).
- As \(x\to-\infty\), \(R(x)\to +\infty\) (left end up).
Exponential & logarithmic equations and properties
Exponentials and logs are inverse operations. For exponential equations, rewrite so the bases match, then set the exponents equal. For log equations, if the same log appears on both sides, drop the logs and set the inside expressions equal (but remember: log inputs must be positive).
Useful facts: \(e^{\ln M}=M\) and \(\ln(e^k)=k\).
Example: One-to-one (exponential)
Solve \(e^{\,3x-7}=e^{-1}\).
Solution
\(3x-7=-1\Rightarrow 3x=6\Rightarrow x=2\).
Example: One-to-one with a quadratic exponent
Solve \(e^{\,(x+2)^{2}-9}=e^{\,0}\).
Solution
\((x+2)^{2}-9=0\Rightarrow (x+2)^{2}=9\Rightarrow x+2=\pm 3 \Rightarrow x=1,-5\).
Example: Evaluate/simplify logs
Compute: (a) \(\log_{3}\!\big(\tfrac{1}{27}\big)\), (b) \(\log_{10}(1000)\), (c) \(e^{\ln(5x-1)}\).
Solution
(a) \(-3\); (b) \(3\); (c) \(5x-1\) (for \(5x-1>0\)).
Example: Expand with log laws
Expand completely: \(\log_{2}\!\Big(\dfrac{x^{3}y^{5}}{z^{2}}\Big)\).
Solution
\(3\log_{2}x+5\log_{2}y-2\log_{2}z\).
Example: Solve a log equation
Solve \(\ln(9x+6)=\ln(60)\).
Solution
\(9x+6=60\Rightarrow 9x=54\Rightarrow x=6\) (valid since \(9x+6>0\)).
Trigonometry: unit circle & graphing
Use the unit circle to compute exact values of \(\sin t\) and \(\cos t\): a point at angle \(t\) has coordinates \((\cos t,\sin t)\). From there, you can compute \(\tan t\), \(\sec t\), and \(\csc t\). Use the quadrant to decide signs.
For graphs, write the sine or cosine function as \[y=A\sin(B(x-C))+D \quad \mbox{ or } \quad y=A\cos(B(x-C))+D\] The amplitude \(=|A|\), period \(=2\pi/|B|\), midline \(y=D\), and the graph shifts right by \(C\). For tangent, \(y=A\tan(B(x-C))+D\) has period \(\pi/|B|\) (no amplitude), midline \(y=D\), and vertical asymptotes at \(x=C\pm \frac{\pi}{2|B|}+k\frac{\pi}{|B|}\).
Example: Six trig functions at a special angle
Evaluate all six trig functions at \(t=\dfrac{7\pi}{6}\).
Solution
\(\sin t=-\tfrac12,\ \cos t=-\tfrac{\sqrt{3}}{2},\ \tan t=\tfrac{\sqrt{3}}{3}\).
\(\csc t=-2,\ \sec t=-\dfrac{2\sqrt{3}}{3},\ \cot t=\sqrt{3}\).
Example: Amplitude & period (cosine)
For \(y=-3\cos(4\pi x)\): find the amplitude and period; describe the graph.
Solution
Amplitude \(3\). Period \(= \dfrac{2\pi}{4\pi}=\tfrac12\). Reflection across midline \(y=0\) due to negative coefficient.
Example: Amplitude, period, vertical shift (sine)
For \(y=1-\tfrac12\sin(\pi x)\): find amplitude, period, and midline.
Solution
Amplitude \(\tfrac12\). Period \(= \dfrac{2\pi}{\pi}=2\). Vertical shift \(+1\); midline \(y=1\).
Example: Tangent period and asymptotes
For \(y=-2\tan\!\big(3(x-\tfrac{\pi}{4})\big)\), find the period and give equations of the first two vertical asymptotes near \(x=\tfrac{\pi}{4}\).
Solution
Period \(=\dfrac{\pi}{3}\).
Asymptotes when \(3(x-\tfrac{\pi}{4})=\pm\tfrac{\pi}{2}+k\pi\Rightarrow x=\tfrac{\pi}{4}\pm \tfrac{\pi}{6}+k\tfrac{\pi}{3}\).
Nearest: \(x=\tfrac{\pi}{12}\) and \(x=\tfrac{5\pi}{12}\).