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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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Most popular questions from this chapter

Stopping Distance When the driver of a vehicle observes an impediment, the total stopping distance involves both the reaction distance \(R\) (the distance the vehicle travels while the driver moves his or her foot to the brake pedal) and the braking distance \(B\) (the distance the vehicle travels once the brakes are applied). For a car traveling at a speed of \(v\) miles per hour, the reaction distance \(R\), in feet, can be estimated by \(R(v)=2.2 v .\) Suppose that the braking distance \(B,\) in feet, for a car is given by \(B(v)=0.05 v^{2}+0.4 v-15\) (a) Find the stopping distance function $$ D(v)=R(v)+B(v) $$ (b) Find the stopping distance if the car is traveling at a speed of \(60 \mathrm{mph}\). (c) Interpret \(D(60)\)

A right triangle has one vertex on the graph of \(y=9-x^{2}, x>0,\) at \((x, y),\) another at the origin, and the third on the positive \(x\) -axis at \((x, 0) .\) Express the area \(A\) of the triangle as a function of \(x\).

Show that a constant function \(f(x)=b\) has an average rate of change of \(0 .\) Compute the average rate of change of \(y=\sqrt{4-x^{2}}\) on the interval \([-2,2] .\) Explain how this can happen.

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Multiple Choice A function that is continuous on the interval ________________ is guaranteed to have both an absolute maximum and an absolute minimum. (a) \((a, b)\) (b) \((a, b]\) (c) \([a, b)\) (d) \([a, b]\)
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