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