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:
\(\text2x\)
\(x^2\)
\(\text2x^2\)
\(\textx^2\)
12.2: Equations and Their Graphs

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:
\(x^2+7\)
\(\text3^2+7\)
\(\text9+7\)
2
Lin:
\(x^2+7\)
\((\text3)^2+7\)
\(9+7\)
16

Evaluate each expression when \(x\) is 4:
 \(x^2\)
 \(\frac12 x^2\)
 \(\text\frac18 x^2\)
 \(\textx^28\)

Using graphing technology, graph \(y = x\). Then, experiment with the following changes to the function. Record your observations (include sketches, if helpful).
 Adding different constant terms to \(x\) (for example: \(x + 4\), \(x  3\)).
 Multiplying \(x\) by different positive coefficients greater than 1 (for example: \(6x, 2.5x\)).
 Multiplying \(x\) by different positive coefficients between 0 and 1 (for example: \(0.25x, 0.1x\)).
 Multiplying \(x\) by negative coefficients (for example: \(\text9x, \text4x\)).

Use your observations to sketch these functions on the coordinate plane, which currently shows \(y=x\).
 \(y =\text0.5x + 2.1\)

\(y = 2.1x  0.5\)
12.3: Match the Graphs
 Evaluate each expression when \(x\) is 3.
 \(x^2\)
 \(\textx^2\)
 \(x^2+20\)
 \(\textx^2+20\)
 For each graph, come up with an equation that the graph could represent. Verify your equation using graphing technology.