MAT137 - Week 10: Graphing Quadratics (Parabolas)

Welcome to Week 10 of MAT137! This week, we're focusing on graphing quadratic functions, also known as parabolas. Understanding parabolas is crucial for various applications in mathematics and beyond. We'll be covering sections 7.1 to 7.3, building upon our previous knowledge of functions and equations.

Key Concepts We'll Cover:

  • The Standard Form of a Quadratic Function: $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a \neq 0$.
  • The Vertex Form of a Quadratic Function: $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This form is incredibly useful for quickly identifying the vertex.
  • Finding the Vertex: The vertex of the parabola can be found using the formula $h = -\frac{b}{2a}$ for the x-coordinate, and then substituting this value back into the original equation to find the y-coordinate, $k = f(h)$. Alternatively, when in vertex form, it is easy to see the vertex coordinates $(h,k)$.
  • Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation of the axis of symmetry is $x = h$.
  • Intercepts:
    • Y-intercept: The point where the parabola intersects the y-axis. To find it, set $x = 0$ in the quadratic equation and solve for $y$.
    • X-intercept(s): The point(s) where the parabola intersects the x-axis. These are also known as the roots or zeros of the quadratic function. To find them, set $f(x) = 0$ and solve for $x$. You can use factoring, the quadratic formula ($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$), or completing the square to find the x-intercepts.
  • The Discriminant: The discriminant, $b^2 - 4ac$, tells us about the nature of the roots:
    • If $b^2 - 4ac > 0$, there are two distinct real roots (two x-intercepts).
    • If $b^2 - 4ac = 0$, there is one real root (the vertex touches the x-axis).
    • If $b^2 - 4ac < 0$, there are no real roots (the parabola does not intersect the x-axis).

Graphing a Parabola: A Step-by-Step Approach

  1. Find the Vertex: Determine the coordinates $(h, k)$ of the vertex.
  2. Find the Axis of Symmetry: Draw the vertical line $x = h$.
  3. Find the Y-intercept: Set $x = 0$ and solve for $y$. Plot the point.
  4. Find the X-intercept(s): Set $y = 0$ and solve for $x$. Plot the point(s). (Remember the discriminant to determine the number of possible intercepts)
  5. Plot Additional Points (if needed): To get a more accurate graph, plot a few additional points on either side of the vertex. Symmetry is your friend! If you know a point exists on one side, you know one exists on the other at the same y-value.
  6. Draw the Parabola: Sketch a smooth curve through the points, ensuring it's symmetrical about the axis of symmetry.

Example:

Let's graph the quadratic function $f(x) = x^2 - 4x + 3$.

  1. Vertex: $h = -\frac{-4}{2(1)} = 2$. $k = f(2) = (2)^2 - 4(2) + 3 = -1$. Vertex: $(2, -1)$.
  2. Axis of Symmetry: $x = 2$.
  3. Y-intercept: $f(0) = 0^2 - 4(0) + 3 = 3$. Y-intercept: $(0, 3)$.
  4. X-intercepts: $x^2 - 4x + 3 = 0 \implies (x - 1)(x - 3) = 0 \implies x = 1, 3$. X-intercepts: $(1, 0)$ and $(3, 0)$.

By plotting these points and drawing a smooth curve, you'll have a graph of the parabola $f(x) = x^2 - 4x + 3$.

Remember to review your class notes from 11-04-08, specifically sections 7.1 to 7.3, for further examples and explanations. And don't forget to tackle the weekly quiz to test your understanding!

Good luck, and have fun graphing! Remember, practice makes perfect. The more you graph, the more comfortable you'll become with these concepts. If you have any questions, don't hesitate to ask!