9-5-2023 Class Notes

Welcome to a recap of Professor Baker's Math Class from September 5th, 2023! Today's session was packed with essential concepts in graphing and function analysis. Let's dive into the key topics covered:

Topics Covered

  • Parabolas
    • Finding the vertex, intercepts, and axis of symmetry from the graph of a parabola. Remember, the vertex form of a parabola is given by $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex.
    • Understanding how the leading coefficient 'a' affects the shape of a 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.
    • Graphing parabolas in the form $y = ax^2$, $y = ax^2 + c$, and $y = (x-h)^2 + k$.
    • Translating parabolas: One and two-step translations. For example, $y = (x-2)^2 + 3$ shifts the standard parabola $y=x^2$ two units to the right and three units up.
  • Symmetry
    • Determining if graphs have symmetry with respect to the x-axis, y-axis, or origin.
    • Testing an equation for symmetry about the axes and origin.
      • X-axis symmetry: Replace $y$ with $-y$. If the equation remains the same, it's symmetric about the x-axis.
      • Y-axis symmetry: Replace $x$ with $-x$. If the equation remains the same, it's symmetric about the y-axis.
      • Origin symmetry: Replace $x$ with $-x$ and $y$ with $-y$. If the equation remains the same, it's symmetric about the origin.
  • Local and Absolute Extrema
    • Finding local maxima and minima of a function given the graph. Local extrema are the highest or lowest points in a specific interval.
    • Finding the absolute maximum and minimum of a function given the graph. Absolute extrema are the highest or lowest points over the entire domain of the function.
  • Function Analysis
    • Finding values and intervals where the graph of a function is zero, positive, or negative. This involves identifying the x-intercepts and analyzing the graph's behavior between these intercepts.
  • Graphing Functions
    • Graphing an absolute value equation of the form $y = A|x|$.
    • Graphing a cubic function of the form $y = ax^3$.
    • Graphing a square root function: Problem type 1, such as $f(x) = \sqrt{x-1}$.
    • Graphing a cube root function.
  • Transformations of Functions
    • Translating the graph of a function: One step and two steps.
    • Writing an equation for a function after a vertical translation. For example, shifting $f(x) = 5x^2 - 1$ down by 7 units results in $f(x) = 5x^2 - 8$.
    • Transforming the graph of a function by reflecting over an axis.
    • Transforming the graph of a function by shrinking or stretching.
    • Transforming the graph of a function using more than one transformation.
    • Transforming the graph of a quadratic, cubic, square root, or absolute value function.

Keep practicing these concepts, and you'll become a graphing pro in no time! Remember, understanding the underlying principles is key to mastering these skills. Good luck, and see you in the next class!