Graphing Equations and Intercepts

Graphing Equations

Part 1: Graphing Equations

• This section focuses on sketching graphs of equations.

• The method used is standard but not precise; calculus is needed for exact graphs.

Steps to Sketch Graphs of Equations:

1. Solve for y in terms of x or x in terms of y, if possible.

2. Pick arbitrary values for x (or y if you solved for x in terms of y).

   • Plug the chosen values into the equation to find corresponding values of y (or x).

   • This creates a table of points on the graph.

3. Plot the obtained points.

4. Join the points to sketch the curve.

• This method works best for simple equations.

Example: Sketch the Graph of y - x^2 = 1

1. Solve for y in terms of x:
   
   y = x^2 + 1

2. Form a table of points:

   • Choose arbitrary values for x (e.g., -2, -1, 0, 1, 2).

   • Calculate the corresponding values of y:

   • If x = 0, then y = 0^2 + 1 = 1

   • If x = 1, then y = 1^2 + 1 = 2

   • If x = 2, then y = 2^2 + 1 = 5

   • If x = -1, then y = (-1)^2 + 1 = 2

   • If x = -2, then y = (-2)^2 + 1 = 5

3. Plot the points (0, 1), (1, 2), (2, 5), (-1, 2), and (-2, 5) on a coordinate plane.

4. Join the points with a smooth curve.
   
   This graph is a parabola.

Intercepts

• Y-intercepts: Points where the graph intersects the y-axis.

  • To find the y-intercept(s), set x = 0 and solve for y.

  • Y-intercepts are of the form (0, y).

• X-intercepts: Points where the graph intersects the x-axis.

  • To find the x-intercept(s), set y = 0 and solve for x.

  • X-intercepts are of the form (x, 0).

Symmetry

• Understanding symmetry can simplify graphing.

• Types of Symmetry:

  • With respect to the y-axis (Even Function): If replacing x with -x results in the original equation, the graph is symmetric with respect to the y-axis.

  • Mathematically: f(-x) = f(x).

  • Example: y = x^2

  • With respect to the x-axis: If replacing y with -y results in the original equation, the graph is symmetric with respect to the x-axis.

  • Mathematically: -y = f(x).

  • Example: x = y^2

  • With respect to the origin (Odd Function): If replacing both x with -x and y with -y results in the original equation, the graph is symmetric with respect to the origin.

  • Mathematically: f(-x) = -f(x).

  • Example: y = x^3

Asymptotes

• Asymptotes are lines that the graph of a function approaches but does not touch.

• Types of Asymptotes:

  • Horizontal Asymptote: A horizontal line that the graph approaches as x tends