Summary of How To Graph Equations - Linear, Quadratic, Cubic, Radical, & Rational Functions

Main Ideas and Concepts

Methods to graph linear equations:
- Using a table: Choose values for \(x\), compute \(y\), and plot points.
- Slope-intercept form \(y = mx + b\):
  - Plot the y-intercept \((0, b)\).
  - Use the slope \(m = \frac{\text{rise}}{\text{run}}\) to find additional points.
- Standard form \(Ax + By = C\):
  - Find x-intercept by setting \(y=0\).
  - Find y-intercept by setting \(x=0\).
  - Plot these intercepts and draw the line.
- Special cases:
  - \(y = c\) is a horizontal line.
  - \(x = c\) is a vertical line.
Graphing linear inequalities:
- Convert to Slope-intercept form.
- Use dashed lines for strict inequalities (\(>\), \(<\)) and solid lines for inclusive inequalities (\(\geq\), \(\leq\)).
- Shade the region above or below the line depending on the inequality.
- For systems of inequalities, shade the overlapping region.

Standard form: \(y = ax^2 + bx + c\)
- Find vertex using \(x = -\frac{b}{2a}\).
- Create a table of points around the vertex (symmetrical about the vertex).
- Plot points and draw the parabola.
- Axis of symmetry is \(x = -\frac{b}{2a}\).
- Minimum or maximum value is the y-coordinate of the vertex depending on the sign of \(a\).
Vertex form: \(y = a(x - h)^2 + k\)
- Vertex is \((h, k)\).
- Graph by plotting vertex and using the pattern of squares to find other points:
  - From vertex, move 1 unit right/left and go up/down \(a \times 1^2\).
  - Move 2 units right/left and go up/down \(a \times 2^2\), etc.
- Completing the square can convert Standard form to Vertex form.

Parent function: \(y = |x|\) is a V-shape with slope 1 on both sides.
Transformations:
- Horizontal shift inside the absolute value.
- Vertical shift outside the absolute value.
- Negative sign outside flips the graph upside down.
Use vertex and slope to plot points symmetrically.

Parent function: \(y = x^3\) has an S-shape passing through the origin.
Transformations include horizontal and vertical shifts.
Plot points using values of \(x^3\) (e.g., \(1^3=1\), \(2^3=8\), \((-1)^3 = -1\)).
Negative coefficients reflect the graph vertically.

Parent function: \(y = \sqrt{x}\) starts at origin and increases slowly.
Transformations:
- Horizontal and vertical shifts.
- Negative outside reflects over x-axis.
- Negative inside reflects over y-axis.
Use perfect squares to select points for easy calculation.
Domain and range:
- Domain starts at the horizontal shift point and goes to infinity.
- Range starts at vertical shift and goes to infinity.
For cube root and other odd roots:
- Graph is symmetric about the origin.
- Similar transformations and plotting points based on cube roots of numbers.

Parent functions:
- \(y = \frac{1}{x}\) has vertical and horizontal asymptotes.
- \(y = \frac{1}{x^2}\) is symmetric about y-axis with both branches above x-axis.
Steps:
- Find vertical asymptotes by setting denominator = 0.
- Find horizontal asymptotes based on degrees of numerator and denominator.
- Plot points near asymptotes.
- Recognize holes where factors cancel.
For more complex rational functions:
- Factor numerator and denominator.
- Identify holes, vertical and horizontal asymptotes.
- For slant (oblique) asymptotes (numerator degree = denominator degree + 1), perform polynomial division to find the asymptote.
- Plot slant asymptote as a linear function and sketch the graph accordingly.