The Graph of a Function

  1. For each of the following functions defined by a formula, plot the graph of the function in Desmos. Change the viewport so that it nicely captures all the important features of the graph: \(x\)- and \(y\)-intercepts, and local maximum and minimums (high and low points). In Desmos you can do this by “pinching” to zoom; pinching diagonally will preserve the viewport’s aspect ratio, whereas pinching along the \(x\)- or \(y\)-axis will zoom only in that dimension. The left, right, top, and bottom bounds of the viewport can be set manually by pressing the l’il “wrench” icon. Also infer the domain and range of the each function from its graph. Before plotting each graph in Desmos, clear the previous graph and press the l’il “home” icon to reset the viewport.

    \(\displaystyle f(x) = 3-\tfrac{1}{2}x\)
    \(\displaystyle g(x) = 2x+17\)
    \(\displaystyle h(x) = \tfrac{2}{11}x+3\)
    \(\displaystyle j(x) = x^2+4x-4\)
    \(\displaystyle k(x) = \tfrac{1}{7}x^2-x-7\)
    \(\displaystyle \ell(x) = -21+31x-11x^2+x^3\)
    \(\displaystyle m(x) = x^3+6x^2+5x+2\)
    \(\displaystyle n(x) = \tfrac{1}{9}x^3+x^2-2x\)
    \(\displaystyle \mathcal{o}(x) = 2\sqrt{3-x}\)
    \(\displaystyle p(x) = \sqrt{7}\)
    \(\displaystyle q(x) = \sqrt{x^2-1}\)
    \(\displaystyle \alpha(x) = \frac{1}{2-x}\)
    \(\displaystyle \beta(x) = \frac{x^2+1}{2x-3}\)
    \(\displaystyle \gamma(x) = \frac{7}{\sqrt{x}}\)
    \(\displaystyle \delta(x) = \sqrt{\frac{2}{x-7}}\)

  2. For each of the following tables of \((x,y)\)-pairs plot a scatter plot for the table in Desmos. Press the l’il “magnifying glass” icon near the table to automatically fit the viewport to the scatter plot. Decide if the scatter plot is perfectly linear or somewhat linear or not even close to linear (note that this is a subjective determination). Before plotting each scatter plot in Desmos, clear the previous scatter plot and press the l’il home button to reset the viewport.

    \(x\) \(y\)
    0 3
    1 5
    2 7
    \(x\) \(y\)
    1 -1
    2 0
    3 2
    \(x\) \(y\)
    -1 0
    2 -3
    5 3
    \(x\) \(y\)
    9 -10
    1 4
    -6 12
    \(x\) \(y\)
    9 45
    23 0
    67 23
    \(x\) \(y\)
    2 21
    5 33
    9 49
    \(x\) \(y\)
    1 -6
    17 1
    24 5
    \(x\) 1 2 3 4 5 6
    \(y\) 0 2 1 3 7 6
    \(x\) 0 10 20 30 40 50 60 70 80
    \(y\) -15 6 21 41 58 69 88 105 121