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Chapter 3: Problem 79

Suppose (1,3) is a point on the graph of \(y=f(x)\) (a) What point is on the graph of \(y=f(x+3)-5 ?\) (b) What point is on the graph of \(y=-2 f(x-2)+1 ?\) (c) What point is on the graph of \(y=f(2 x+3) ?\)

Short Answer

Expert verified
(-2, -2), (3, -5), (-1, 3)

Step by step solution

01

- Understand the Original Point

Given point (1,3) is on the graph of the function \(y=f(x)\). This means when \(x=1\), \(y=f(1)=3\).
02

- Solve for \(y=f(x+3)-5\)

To find the new point for the transformed function \(y=f(x+3)-5\), first note that the original x-value is 1: we need to solve for \(x+3=1\). So, \(x = 1 - 3 = -2\). Then, substitute this x-value into the transformed function: \(y = f(-2 + 3) - 5 = f(1) - 5\). Since \(f(1) = 3\), it follows that \(y = 3 - 5 = -2\). So, the new point is (-2, -2).
03

- Solve for \(y=-2 f(x-2)+1\)

To find the point on the graph of \(y=-2 f(x-2)+1\), start with \(x-2=1\). So, \(x=1+2=3\). Then, substitute this x-value into the transformed function: \(y = -2 f(3-2) + 1 = -2 f(1) + 1\). Since \(f(1)=3\), it follows that \(y = -2 \times 3 + 1 = -6 + 1 = -5\). So, the new point is (3, -5).
04

- Solve for \(y=f(2x+3)\)

To find the point on the graph of \(y=f(2x+3)\), set \(2x+3=1\). Solving for x gives \(2x = 1 - 3 = -2\), so \(x = -1\). Then, substitute this x-value into the transformed function: \(y=f(2(-1)+3)=f(-2+3)=f(1)\). Since \(f(1)=3\), the new point is (-1, 3).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Functions
Graphing functions is essential in algebra. Functions show a relationship between two variables, typically x and y. We plot these relationships on a coordinate plane where the x-axis represents the input values and the y-axis represents the output values. For example, the point (1,3) on the graph of the function \(y=f(x)\) means that when \(x=1\), the function's output is \(y=3\).
Understanding the coordination of these points helps us visualize and analyze the behavior of the function.
Function Translation
Function translation shifts the graph of a function horizontally or vertically. When given \(y=f(x+3)-5\), the term \(x+3\) translates the graph 3 units to the left and \(-5\) translates it down by 5 units. For instance, with an original point of (1,3), finding the new point involves understanding these shifts:
  • Horizontally: Solve for \(x+3=1\) to get \(x=-2\).
  • Vertically: Substitute \(-2\) into the function to find \(y=f(1)-5\), which gives \(y=-2\) assuming \(f(1)=3\).
The transformed point is (-2, -2). This translation helps in visualizing and comparing function movements on a graph.
Function Scaling
Function scaling involves stretching or compressing the graph. For instance, with \(y=-2f(x-2)+1\), the factor -2 scales the function by 2 and reflects it across the x-axis. Here, \(x-2\) shifts the graph 2 units to the right, and the +1 vertically shifts it up by 1 unit:
  • Horizontally: Solve for \(x-2=1\) gives \(x=3\).
  • Scaling and translating: Substitute \(x=3\) into the function \(-2f(1)+1\). With \(f(1)=3\), this gives \(y=-5\).
Therefore, the point transformed is (3, -5). Understanding scaling is critical for interpreting modifications in graph shapes and directions.
Solving Equations
Solving equations is about finding the value of variable x that satisfies the equation. For transformation like \(y=f(2x+3)\), the process entails:
  • Equalizing: Set \(2x+3=1\) and solve for \(x\). This results in \(x=-1\).
  • Substitution: Use \(x=-1\) in the function \(f(2(-1)+3)\) to find \(y=3\) assuming \(f(1)=3\).
Here, the new point is (-1, 3). This method shows how we determine shifts and new coordinates for modified function graphs. It's vital for accurately analyzing and understanding transformations and applying them to real-life problems.

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