Work each problem. Find a function g(x)=ax+b whose graph can be obtained by translating the graph of ƒ(x)=2x+5 up 2 units and to the left 3 units.

Verified step by step guidance

1. Start with the original function: \(f(x) = 2x + 5\).
2. Recall that translating a graph up by 2 units means adding 2 to the entire function, so the function becomes \(f(x) + 2\).
3. Translating the graph to the left by 3 units means replacing \(x\) with \((x + 3)\) in the function, so the function becomes \(f(x + 3)\).
4. Apply the horizontal translation first: replace \(x\) with \((x + 3)\) in \(f(x)\) to get \(f(x + 3) = 2(x + 3) + 5\).
5. Then apply the vertical translation by adding 2: \(g(x) = f(x + 3) + 2 = 2(x + 3) + 5 + 2\). Simplify this expression to write \(g(x)\) in the form \(ax + b\).

Key Concepts

Function Translation

Function translation involves shifting the graph of a function horizontally or vertically without changing its shape. Moving a graph up or down adds or subtracts a constant to the function's output, while moving left or right adjusts the input variable inside the function.

Horizontal Translation and Input Adjustment

Translating a graph to the left by a certain number of units means replacing the input variable x with (x + h), where h is the number of units moved left. This shifts the graph horizontally and affects the function's formula accordingly.

Linear Function Form and Parameters

A linear function has the form g(x) = ax + b, where a is the slope and b is the y-intercept. Understanding how translations affect these parameters helps in finding the new function after shifting the original graph.