Graphing Quadratics with Vertex Form

Welcome to Professor Baker's Math Class! Today, we're diving into the world of quadratic functions and exploring how to graph them effectively using the vertex form. Understanding vertex form can simplify the graphing process and provide valuable insights into the behavior of quadratic equations.

What is Vertex Form?

The vertex form of a quadratic function is given by:

$$f(x) = a(x - h)^2 + k$$

Where:

  • $a$ determines the direction and stretch of the parabola. If $a > 0$, the parabola opens upwards; if $a < 0$, it opens downwards. The larger the absolute value of $a$, the narrower the parabola.
  • $(h, k)$ represents the vertex of the parabola. The vertex is the point where the parabola changes direction (minimum or maximum).
  • $h$ represents a horizontal translation.
  • $k$ represents a vertical translation.

Key Steps for Graphing

  1. Identify the Vertex: From the vertex form $f(x) = a(x - h)^2 + k$, the vertex is simply $(h, k)$. Remember that the $h$ value is the opposite sign of what appears inside the parentheses.
  2. Determine the Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation for the axis of symmetry is $x = h$.
  3. Find Additional Points: Choose a few $x$-values on either side of the vertex and calculate the corresponding $f(x)$ values. This will give you additional points to plot and help you draw an accurate parabola. Symmetry can be used to quickly find matching points on the other side of the axis of symmetry.
  4. Plot the Points and Sketch the Graph: Plot the vertex, the additional points you calculated, and draw a smooth curve to create the parabola.

Example

Let's graph the quadratic function $f(x) = 2(x + 1)^2 - 2$.

  1. Identify the Vertex: The vertex is $(-1, -2)$.
  2. Axis of Symmetry: The axis of symmetry is $x = -1$.
  3. Find Additional Points:
    • If $x = -3$, then $f(-3) = 2(-3 + 1)^2 - 2 = 2(4) - 2 = 6$.
    • If $x = -2$, then $f(-2) = 2(-2 + 1)^2 - 2 = 2(1) - 2 = 0$.
    • If $x = 0$, then $f(0) = 2(0 + 1)^2 - 2 = 2(1) - 2 = 0$.
    • If $x = 1$, then $f(1) = 2(1 + 1)^2 - 2 = 2(4) - 2 = 6$.

    So we have the points $(-3, 6)$, $(-2, 0)$, $(0, 0)$, and $(1, 6)$.

  4. Plot and Sketch: Plot the vertex $(-1, -2)$ and the additional points. Draw a smooth curve to complete the parabola.

Discussion Question

What is the hardest part of graphing quadratics in the vertex form and why? Bonus points for finding an easier way to complete this step!

Core Math Worksheet

Don't forget to complete the Core Math Worksheet for 2-1 to practice what you've learned. Good luck!