Lesson 12

Changing the Equation

  • Let's look at quadratics with negative inputs.

12.1: Math Talk: A Negative Input

Evaluate each expression when \(x\) is -5:

\(\text-2x\)

\(x^2\)

\(\text-2x^2\)

\(\text-x^2\)

12.2: Equations and Their Graphs

  1. Two students are evaluating \(x^2+7\) when \(x\) is -3. Here is their work. Do you agree with either of them? Explain your reasoning. 

    Tyler:

    \(x^2+7\)

    \(\text-3^2+7\)

    \(\text-9+7\)

    -2

    Lin:

    \(x^2+7\)

    \((\text-3)^2+7\)

    \(9+7\)

    16

  2. Evaluate each expression when \(x\) is -4:

    1. \(x^2\)
    2. \(\frac12 x^2\)
    3. \(\text-\frac18 x^2\)
    4. \(\text-x^2-8\)
  3. Using graphing technology, graph \(y = x\). Then, experiment with the following changes to the function. Record your observations (include sketches, if helpful).

    1. Adding different constant terms to \(x\) (for example: \(x + 4\), \(x - 3\)).
    2. Multiplying \(x\) by different positive coefficients greater than 1 (for example: \(6x, 2.5x\)).
    3. Multiplying \(x\) by different positive coefficients between 0 and 1 (for example: \(0.25x, 0.1x\)).
    4. Multiplying \(x\) by negative coefficients (for example: \(\text-9x, \text-4x\)).
  4. Use your observations to sketch these functions on the coordinate plane, which currently shows \(y=x\)

    1. \(y =\text-0.5x + 2.1\)
    2. \(y = 2.1x - 0.5\)

      graph of y = x
      graph of y = x

12.3: Match the Graphs

  1. Evaluate each expression when \(x\) is -3.
    1. \(x^2\)
    2. \(\text-x^2\)
    3. \(x^2+20\)
    4. \(\text-x^2+20\)
  2. For each graph, come up with an equation that the graph could represent. Verify your equation using graphing technology. 
    graphs A, B, C, D, E, F of 6 lines. 

Summary