College Algebra Math 1314 Essential Formulas

Posted on Aug 14, 2026 in Mathematics

1. Equations & Analytical Domains

  • Radical Equations (\(x – \sqrt{\text{expr}} = c\)):


    Isolate root \(\rightarrow \) square both sides \(\rightarrow \) solve quadratic \(\rightarrow \)

    Always check for extraneous solutions


  • Fractional Exponents (\((ax+b)^{1/3} + c = d\)):

    Isolate exponent term \(\rightarrow \) cube both sides to eliminate the \(1/3\) power \(\rightarrow \) solve for \(x\).

  • Rational Functions (\(\frac{\text{top}}{\text{bottom}}\)):

    Set \(\text{bottom} \neq 0\). Solve to find excluded values.

  • Square Roots (\(\sqrt{\text{inside}}\)):

    Set \(\text{inside} \geq 0\) and solve.

  • Root in Denominator (\(\frac{1}{\sqrt{\text{inside}}}\)):

    Set \(\text{inside} > 0\) and solve.
2. Graphs & Intervals

  • Vertical Line Test:


    If a vertical line hits the graph more than once, it is not a function.

  • Finding \(f(c)\):

    Find \(c\) on the x-axis \(\rightarrow \) go vertically to the graph line \(\rightarrow \) read the y-value.

  • Domain & Range:

    Domain = scan x-axis left–
    To-right. Range = scan y-axis bottom-to-top.

  • Max / Min Points:

    Write as an ordered pair \((x, y)\). Peaks are relative maxima; valleys are relative minima.

  • Behaviors (Inc / Dec / Constant):

    Read left-to-right. Use x-axis values only for boundaries. Always write using open intervals (a, b).
3. Key Algebraic Workflows

  • Even vs. Odd Functions:


    • Even:


      \(f(-x) = f(x)\) (all variable signs stay the same).

    • Odd:

      \(f(-x) = -f(x)\) (all variable signs flip perfectly).

  • Piecewise Functions:


    Look at the constraint boundaries on the right side first \(\rightarrow \) identify the single matching row for your given input value \(\rightarrow \) plug in and solve.

  • The Difference Quotient:

    \(\frac{f(x+h)-f(x)}{h}\)
    1. Substitute \((x+h)\) into every \(x\) slot and expand completely.
    2. Subtract the entire original function (distribute the negative sign).
    3. Simplify terms, factor out \(h\) from the numerator, and cancel out the bottom \(h\).

  • Inverse Functions (\(f^{-1}(x)\)):


    Set \(f(x) = y\) \(\rightarrow \) swap every \(x\) and \(y\) variable \(\rightarrow \) isolate the new \(y\) \(\rightarrow \) rewrite as \(f^{-1}(x)\).
4.
Function Transformations & Base Shapes

  • Shifts:


    \(f(x – c)\) moves Right | \(f(x + c)\) moves Left | \(+d\) moves Up | \(-d\) moves Down.

  • Reflections:

    \(-f(x)\) flips over x-axis | \(f(-x)\) flips over y-axis.

  • Parent Graphs:

    Linear (line) | Absolute Value (V-shape) | Quadratic (U-shape) | Cubic (S-wave) | Square Root (half-arch).
5. Quadratic Properties (\(f(x) = ax^2 + bx + c\))

  • Vertex Point:


    Find \(x = -\frac{b}{2a}\), then plug that answer back in to get \(y = f\left(-\frac{b}{2a}\right)\). Write as \((x, y)\).

  • Axis of Symmetry:

    Always write as a vertical line equation:
    \(x = -\frac{b}{2a}\).

  • Intercepts:

    y-
    intercept is always \((0, c)\). X-intercepts are found via factoring or the quadratic formula:

    \(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\)

College Algebra (Math 1314)
Cheat Sheet — Part 2

1.
Quadratic Functions & Graphs

  • Vertex & Max/Min:


    Find \(x = -\frac{b}{2a}\), then compute \(y = f(x)\).
    • If \(a > 0\), graph opens Up \(\rightarrow \) Minimum value at \(y\).
    • If \(a < 0\), graph opens Down \(\rightarrow \) Maximum value at \(y\).

  • Domain & Range from Vertex \((h, k)\):


    • Domain is always \((-\infty, \infty)\).
    • Range if opening up: \([k, \infty)\) | Range if opening down: \((-\infty, k]\).
2. Polynomial Characteristics & End Behavior

  • Turning Points:


    Maximum number of turning points is always \(\text{degree} – 1\).

  • End Behavior (Leading Coefficient Test):

    • Even Degree:


      Both ends go same way. Positive \(a\) \(\rightarrow \) \((\uparrow, \uparrow)\) | Negative \(a\) \(\rightarrow \) \((\downarrow, \downarrow)\).

    • Odd Degree:

      Ends go opposite ways. Positive \(a\) \(\rightarrow \) \((\downarrow, \uparrow)\) | Negative \(a\) \(\rightarrow \) \((\uparrow, \downarrow)\).

  • Multiplicity of Zeros:


    • Odd Multiplicity:


      Graph crosses the x-axis at that zero.

    • Even Multiplicity:

      Graph touches and turns around at that zero.
3. Dividing Polynomials & Finding Zeros

  • Synthetic Division Checklist:


    Only divide by \((x – c)\). If dividing by \((x + 5)\), use \(-5\) in the outside box.
    Never forget a zero placeholder for any missing power terms.

  • Remainder Theorem:

    The remainder from dividing \(f(x)\) by \((x – c)\) is exactly equal to evaluating \(f(c)\).

  • Rational Zero Theorem (\(p/q\)):

    List all possible rational roots using:
    \(\frac{p}{q}=\frac{\text{Factors\ of\ constant\ term\ (last\ number)}}{\text{Factors\ of\ leading\ coefficient\ (first\ number)}}\)
4. Rational Functions (\(f(x) = \frac{\text{Numerator}}{\text{Denominator}}\))

  • Vertical Asymptotes (VA):


    Simplify the fraction completely, set \(\text{Denominator} = 0\), and solve for \(x\).

  • Horizontal Asymptotes (HA):

    Compare the highest degree of the top (\(n\)) vs. Bottom (\(m\)):

    • Top heavy (\(n > m\)):


      No horizontal asymptote.

    • Bottom heavy (\(n < m\)):

      HA is always the line \(y = 0\).

    • Degrees equal (\(n = m\)):

      HA is the line \(y = \frac{\text{leading coefficient of top}}{\text{leading coefficient of bottom}}\).
5. Inequalities & Testing Regions

  • Polynomial Inequalities (\(f(x) > 0\) or \(f(x) < 0\)):


    1. Set the expression equal to zero and solve to find your critical points.
    2. Plot critical points on a number line to divide it into test regions.
    3. Pick a test number inside each region, plug it into the expression, and check the sign (+ or -).
    4. Write the matching positive or negative regions in interval notation.
6. Exponentials & Logarithms

  • Graphs of \(f(x) = b^x\):


    Always have a horizontal asymptote at \(y = 0\). Domain is \((-\infty, \infty)\) and Range is \((0, \infty)\).

  • Form Conversion Rule:


    \(\log _{b}(x)=y\quad \iff \quad b^{y}=x\)

  • Compound Interest Formulas:

    • Compounded \(n\) times per year:


      \(A = P\left(1 + \frac{r}{n}\right)^{nt}\)

    • Compounded continuously:

      \(A = Pe^{rt}\)

  • College Algebra (Math 1314) Cheat Sheet — Part 3

    1. Logarithm Properties & Evaluation

    • Basic Rules:


      \(\log_b(1) = 0\) | \(\log_b(b) = 1\) | \(\log_b(b^x) = x\) | \(\ln(e^x) = x\)

    • Roots & Fractions:

      Rewrite roots as fractional exponents \(\sqrt[n]{x} = x^{1/n}\) and fractions as negative exponents \(\frac{1}{x^n} = x^{-n}\) before evaluating.

    • Expanding vs. Condensing:

      • Product Rule:


        \(\log_b(xy) = \log_b(x) + \log_b(y)\)

      • Quotient Rule:

        \(\log_b(\frac{x}{y}) = \log_b(x) – \log_b(y)\)

      • Power Rule:

        \(\log_b(x^k) = k \cdot \log_b(x)\) (Move exponents to the front)
    2. Solving Exponential & Logarithmic Equations

    • Same Base Method:


      If \(b^x = b^y\), then set exponents equal: \(x = y\). Keep bases prime (e.G., rewrite \(27\) as \(3^{3}\), or \(\frac{1}{e^{8}}\) as \(e^{-8}\)).

    • Different Bases (\(b^x = a\)):

      Take the natural log (\(\ln \)) of both sides \(\rightarrow \) drop the exponent down using the power rule \(\rightarrow \) isolate \(x\).

    • Single Log Equations (\(\log_b(x) = y\)):

      Convert directly to exponential form: \(b^y = x\).

    • Multiple Log Equations:

      Use condensing rules to get a single log on each side \(\rightarrow \) drop the logs \(\rightarrow \) solve the remaining algebra.

      Reject any solution that makes the inside of an original log negative or zero

    3. Systems of Equations & Matrices

    • Checking Solutions:


      Plug the ordered pair \((x, y)\) into both equations. It must make both statements true.

    • Substitution Method:

      Isolate one variable in one equation \(\rightarrow \) plug that expression into the other equation \(\rightarrow \) solve.

    • Addition (Elimination) Method:

      Multiply one or both equations by numbers that cause one variable’s coefficients to be exact opposites \(\rightarrow \) add equations to eliminate that variable.

    • Matrix Scalar Operations:

      To find \(-3A – 2B\), multiply every element in matrix \(A\) by \(-3\), multiply every element in matrix \(B\) by \(-2\), and add the corresponding positions.

    • Matrix Multiplication (\(A \times B\)):

      • Dimension Rule:


        Matrix multiplication is only possible if the number of columns in A equals the number of rows in B.

      • Procedure:

        Multiply the items in row \(i\) of matrix \(A\) by the items in column \(j\) of matrix \(B\) and add them together to find position \((i,j)\).
    4. Determinants & Cramer’s Rule
    • For a \(2 \times 2\) system of equations (\(ax + by = c\) and \(dx + ey = f\)):

    • Determinant \(D\):

      Main matrix coefficients \(\rightarrow \) \(\begin{vmatrix} a & b \\ d & e \end{vmatrix} = ae – bd\)

    • Determinant \(D_{x}\):

      Replace x-column with constants \(\rightarrow \) \(\begin{vmatrix} c & b \\ f & e \end{vmatrix} = ce – bf\)

    • Determinant \(D_{y}\):

      Replace y-column with constants \(\rightarrow \) \(\begin{vmatrix} a & c \\ d & f \end{vmatrix} = af – cd\)

    • Final Answers:

      \(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\)
    5. Sequences & Series

    • Summation Notation (\(\sum_{i=k}^{n} \text{expr}\)):


      Plug in every integer from the bottom index \(k\) up to the top limit \(n\) into the expression, then add all resulting values together.

    • Arithmetic Sequences (Common Difference \(d\)):

      • Formula:


        \(a_n = a_1 + (n – 1)d\)

      • Workflow:

        Find \(d\) by subtracting consecutive terms \(\rightarrow \) plug in \(a_{1}\) and \(d\) \(\rightarrow \) simplify to get the general formula \(\rightarrow \) plug in your target \(n\) to find a specific term.

    • Geometric Sequences (Common Ratio \(r\)):


      • Formula:


        \(a_n = a_1 \cdot r^{n-1}\)

      • Workflow:

        Find \(r\) by dividing consecutive terms (\(\frac{a_{2}}{a_{1}}\)) \(\rightarrow \) plug in \(a_{1}\) and \(r\) to get the general formula \(\rightarrow \) use exponents to find specific terms.