# 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. $$y = 2.1x - 0.5$$

### 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.