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Chapter 2: Problem 9

Given the function \(Q(t)=13-5 t,\) construct a related function whose graph: a. Lies five units above the graph of \(Q(t)\) b. Lies three units below the graph of \(Q(t)\) c. Has the same vertical intercept d. Has the same slope e. Has the same steepness, but the slope is positive

Short Answer

Expert verified
a) 18 - 5t, b) 10 - 5t, c) 13 - 5t, d) 7 - 5t, e) 3 + 5t

Step by step solution

01

Identify the Original Function

The given function is \(Q(t) = 13 - 5t\).
02

Function 5 Units Above

To move the graph up by 5 units, add 5 to the original function: \(Q(t) + 5\). The new function is \(18 - 5t\).
03

Function 3 Units Below

To move the graph down by 3 units, subtract 3 from the original function: \(Q(t) - 3\). The new function is \(10 - 5t\).
04

Same Vertical Intercept

A function that maintains the same vertical intercept \(13\) could be any function of the form \(13 + kt\). Therefore, one example is \(13 - 5t\), which is the original function itself.
05

Same Slope

A function with the same slope \(-5\) could be any function of the form \(k - 5t\). One example is \(7 - 5t\).
06

Same Steepness, Positive Slope

To change the slope to positive but keeping the same steepness (absolute value), change the slope to positive 5: \(k + 5t\). One example is \(3 + 5t\).

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Key Concepts

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

Vertical Shift
Understanding vertical shifts is essential when working with linear functions. A vertical shift moves the graph of a function up or down without changing its shape. For instance, to derive a function whose graph is five units above the given function \(Q(t) = 13 - 5t\), you need to add 5 to the original equation. This results in \(Q(t) + 5 = 18 - 5t\). Similarly, for a function whose graph is three units below \(Q(t)\), you subtract 3 from the original equation, yielding \(Q(t) - 3 = 10 - 5t\).

Key points:
  • Add a constant to the function to shift its graph upwards.
  • Subtract a constant to shift its graph downwards.
Slope
The slope of a linear function is a measure of its steepness and direction. It鈥檚 the coefficient of the variable term in the function. In \(Q(t) = 13 - 5t\), the slope is \(-5\), indicating a downward slant. A function with the same slope \(-5\) could be of the form \((k - 5t)\). For example, \(7 - 5t\) keeps the slope identical to \(Q(t)\).

Key points:
  • The slope is the number multiplying the variable in the linear function.
  • A negative slope results in a line that descends as it moves from left to right.
Graph Transformation
Graph transformations alter the position and shape of the function's graph in various ways. A vertical shift, which adds or subtracts a constant, is one form of transformation. Another is changing the slope鈥檚 sign while keeping its magnitude, termed a reflection across the horizontal axis. For example, converting \(-5t\) to \(+5t\) while keeping the same steepness (absolute value) would result in a function like \(3 + 5t\), reflecting the graph upwards.

Key points:
  • Vertical shifts adjust the graph up or down.
  • Altering the slope's sign reflects the graph.
Function Intercepts
The intercepts are the points where the graph crosses the axes. For a function like \(Q(t) = 13 - 5t\), the vertical intercept (where the graph crosses the y-axis) occurs when \(t = 0\), which is \(13\). Functions sharing the same vertical intercept could have different slopes but must pass through this intercept. An example would be \(13 + kt\), where \k\ is any constant. In this exercise, the original function \(13 - 5t\) already fits this criterion.

Key points:
  • The vertical intercept is found by setting the variable to zero and solving for the function value.
  • Different functions can share the same vertical intercept while having different slopes.

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