17） Max malk 5 Thegaphof 4（） =2%ー4+7Istansated using the vector （3）

a.) Describe the effect of the translation on the graph of t.

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b.) Write the equation of the translated graph, giving your answer in the form

y = ax2 + bx + c.

To solve this problem, let's break it down step by step. 

### Problem Breakdown:

1. **Given:**
   - The graph of \( t(x) = 2x^2 - 4x + 7 \).
   - The graph is translated using the vector \( \begin{pmatrix} 3 \\ m \end{pmatrix} \).
   - We need to describe the effect of the translation on the graph of \( t(x) \) and then find the equation of the translated graph.

### Part (a): Effect of the Translation on the Graph

A translation vector \( \begin{pmatrix} 3 \\ m \end{pmatrix} \) means:
- **Horizontal shift by 3 units to the right** (because of the \( +3 \) in the x-component of the vector).
- **Vertical shift by \( m \) units** (because of the \( +m \) in the y-component of the vector).

### Part (b): Write the Equation of the Translated Graph

Given the translation vector \( \begin{pmatrix} 3 \\ m \end{pmatrix} \), the translation affects the graph of \( t(x) = 2x^2 - 4x + 7 \) as follows:

1. **Horizontal Shift:**
   - The horizontal shift affects the \( x \)-values, so we replace \( x \) with \( (x - 3) \) in the original equation.

   Substituting \( x \) with \( (x - 3) \) in \( t(x) \):
   \[
   t(x - 3

Understand how the translation vector affects the quadratic function t(x) = 2x^2 - 4x + 7. Learn about horizontal and vertical shifts in graphs and find the equation of the translated graph. Suitable for high school students studying quadratic functions.

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