# Lesson 16

Graphing from the Vertex Form

- Let’s use vertex form to reason about a graph.

### 16.1: Math Talk: When $x$ Is -7

Evaluate each expression when \(x\) is -7:

\(x+4\)

\((x+4)^2\)

\(\text-(x+4)^2\)

\(\text-(x+4)^2+5\)

### 16.2: Four Functions

- Complete the table of values for each function.
\(f(x)=(x-4)^2\)

\(x\) 0 1 2 3 4 5 6 7 \(f(x)\) \(g(x)=\text-(x-4)^2\)

\(x\) 0 1 2 3 4 5 6 7 \(g(x)\) - Use the completed tables to answer these questions:
- What are the coordinates of the vertex of each graph? How can you tell?
- Does the graph of function \(f\) open up or down? How can you tell?
- Does the graph of function \(g\) open up or down? How can you tell?

- Suppose function \(h\) is defined by \(h(x) = (x-4)^2 + 5\) and function \(j\) is defined by \(j(x) = \text-(x-4)^2 + 5\). Make predictions about the graph of each function using the questions here. If you get stuck, try creating a tables of values.
- What are the coordinates of the vertex of the graph of \(h\) and \(j\)?
- Which way—up or down—does the graph of each function open? How do you know?

### 16.3: Four More Functions

Here are some tables of values that represent quadratic functions.

\(x\) | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|

\(t(x)\) | -11 | -2 | 1 | -2 | -11 | -26 | -47 |

\(x\) | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|---|---|

\(u(x)\) | 13 | 4 | 1 | 4 | 13 | 28 | 49 |

\(x\) | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|

\(v(x)\) | 76 | 49 | 28 | 13 | 4 | 1 | 4 |

\(x\) | -4 | -3 | -2 | -1 | 0 | 1 | 2 |
---|---|---|---|---|---|---|---|

\(w(x)\) | -47 | -26 | -11 | -2 | 1 | -2 | -11 |

- Make a rough sketch of a graph of each function. Label the vertex of each graph with its coordinates.

- Here are some expressions that define quadratic functions. Match each function \(t\), \(u\), \(v\), and \(w\) with an expression that defines it.
- \(3x^2 + 1\)
- \(\text-3(x-4)^2+1\)
- \(3(x-4)^2 + 1\)
- \(\text-3x^2 + 1\)