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Chapter 8: Problem 17

Given the function \(g(t)\), identify the simplest function \(f(t)\) (linear, power, exponential, or logarithmic) from which \(g(t)\) could have been constructed. Describe the transformations that changed \(f(t)\) to \(g(t)\) a. \(g(t)=\frac{t-1}{2}\) b. \(g(t)=3\left(\frac{1}{2}\right)^{t+4}\) c. \(g(t)=\frac{-7}{t-5}-2\)

Short Answer

Expert verified
a) Linear from \(t\) with horizontal shift and vertical scaling. b) Exponential from \(\left(\frac{1}{2}\right)^t\) with vertical scaling and horizontal shift. c) Rational from \(\frac{1}{t}\) with reflection, scaling, horizontal and vertical shifts.

Step by step solution

01

Identify simplest function for Part (a)

The given function is \(g(t) = \frac{t-1}{2}\). Identify the simplest function (linear in this case) from which this form could be derived. The simplest function is \(f(t) = t\), as it is the base form before any transformations.
02

Describe transformations for Part (a)

Transformation 1: Horizontal shift. Shift the function right by 1 unit, which means \(f(t)\) becomes \(f(t-1)\). Transformation 2: Vertical scaling. Scale the function by a factor of \(\frac{1}{2}\), which means \(f(t-1)\) becomes \(\frac{f(t-1)}{2}\). Combining these, we get \(g(t) = \frac{t-1}{2}\).
03

Identify simplest function for Part (b)

The given function is \(g(t) = 3 \left( \frac{1}{2} \right)^{t+4}\). Identify the simplest function (exponential) from which this form could be derived. The simplest function is \(f(t) = \left( \frac{1}{2} \right)^t\).
04

Describe transformations for Part (b)

Transformation 1: Vertical scaling. Scale the function by a factor of 3, which means \(f(t)\) becomes \(3f(t)\). Transformation 2: Horizontal shift. Shift the function left by 4 units to account for the \(t+4\) term, which makes \(3 \left( \frac{1}{2} \right)^t\) become \(3 \left( \frac{1}{2} \right)^{t+4}\). Combining these, we get \(g(t) = 3 \left( \frac{1}{2} \right)^{t+4}\).
05

Identify simplest function for Part (c)

The given function is \(g(t) = \frac{-7}{t-5} - 2\). Identify the simplest function (rational) from which this form could be derived. The simplest function is \(f(t) = \frac{1}{t}\).
06

Describe transformations for Part (c)

Transformation 1: Reflection and vertical scaling. Reflect and vertically scale the function by \(-7\), changing \(f(t)\) to \(\frac{-7}{t}\). Transformation 2: Horizontal shift. Shift the function right by 5 units, changing \(\frac{-7}{t}\) to \(\frac{-7}{t-5}\). Transformation 3: Vertical shift. Shift the function down by 2 units, making \(\frac{-7}{t-5}\) become \(\frac{-7}{t-5} - 2\). Combining these, we get \(g(t) = \frac{-7}{t-5} - 2\).

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Key Concepts

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

linear transformation
A linear transformation involves changing a function by using linear operations like shifts and scalings. Let's take part (a) of the exercise: the function given is \( g(t) = \frac{t-1}{2} \). We start with the simplest linear function, which is \( f(t) = t \). The transformations applied here are: a horizontal shift to the right by 1 unit, converting \( t \) to \( t-1 \); then a vertical scaling by a factor of \( \frac{1}{2} \), making the transformation complete. Essentially, a linear transformation maintains the basic 'line' essence of the function but adjusts its position and slope. Understanding this helps us to break down and manipulate linear equations effectively.
exponential function
An exponential function changes significantly faster than a linear one. In part (b) of the exercise, the given function is \( g(t)=3\left(\frac{1}{2}\right)^{t+4} \). We start with the simplest exponential function \( f(t)=\left(\frac{1}{2}\right)^t \). The transformation here involves vertical scaling by 3, multiplying the function by 3. This makes the function steeper. The second transformation is a horizontal shift to the left by 4 units, changing \( t \) to \( t+4 \). Exponential functions are unique because of their growth (or decay) patterns - they grow (or shrink) very rapidly, and transformations can significantly alter their behavior.
rational function
Rational functions consist of ratios of polynomials. In part (c) of the exercise, we look at \( g(t) = \frac{-7}{t-5} -2 \). Here, the simplest form of the function is \( f(t) = \frac{1}{t} \). First, a reflection and vertical scaling change this to \( \frac{-7}{t} \). Next, a horizontal shift to the right by 5 units modifies the function to \( \frac{-7}{t-5} \). Finally, a vertical shift downwards by 2 units results in \( \frac{-7}{t-5} - 2 \). Rational functions often have interesting asymptotic behavior that can be shifted and scaled in various ways without changing their essential nature.
horizontal shift
Horizontal shifts move the graph of a function left or right on the coordinate plane. In part (a), the function \( f(t) = t \) undergoes a horizontal shift to the right by 1 unit, becoming \( f(t-1) \). In part (b), the function \( \left(\frac{1}{2}\right)^t \) shifts left by 4 units, resulting in \( \left(\frac{1}{2}\right)^{t+4} \). Lastly, in part (c), \( \frac{-7}{t} \) shifts right by 5 units to become \( \frac{-7}{t-5} \). Knowing how to apply and recognize horizontal shifts helps us understand how the graph and behavior of a function change along the x-axis.
vertical scaling
Vertical scaling changes the amplitude or steepness of the graph. In part (a), \( f(t) = t \) is scaled vertically by \( \frac{1}{2} \), leading to \( \frac{t-1}{2} \). For part (b), \( \left(\frac{1}{2}\right)^t \) is multiplied by 3 to become \( 3\left(\frac{1}{2}\right)^t \). In part (c), \( \frac{1}{t} \) undergoes vertical scaling and reflection by \( -7 \) to become \( \frac{-7}{t} \). Vertical scaling affects how steep or flat the graph presents itself and is important for tuning the function to match specific criteria.

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Most popular questions from this chapter

The wind chill temperature is the apparent temperature caused by the extra cooling from the wind. A rule of thumb for estimating the wind chill temperature for an actual temperature \(t\) that is above \(0^{\circ}\) Fahrenheit is \(W(t)=t-1.5 S_{0}\), where \(S_{0}\) is any given wind speed in miles per hour. a. If the wind speed is 25 mph and the actual temperature is \(10^{\circ} \mathrm{F}\), what is the wind chill temperature? We know how to convert Celsius to Fahrenheit; that is, we can write \(t=F(x),\) where \(F(x)=32+\frac{9}{5} x,\) with \(x\) the number of degrees Celsius and \(F(x)\) the equivalent in degrees Fahrenheit. b. Construct a function that will give the wind chill temperature as a function of degrees Celsius. c. If the wind speed is \(40 \mathrm{mph}\) and the actual temperature is \(-10^{\circ} \mathrm{C},\) what is the wind chill temperature?

Without drawing the graph, list the following parabolas in order, from the narrowest to the broadest. Verify your results with technology. a. \(y=x^{2}+20\) d. \(y=4 x^{2}\) b. \(y=0.5 x^{2}-1\) e. \(y=0.1 x^{2}+2\) c. \(y=\frac{1}{3} x^{2}+x+1\) f. \(y=-2 x^{2}-5 x+4\)

In Exercises \(29-32,\) for each function \(Q\) find \(Q^{-1},\) if it exists. For those functions with inverses, find \(Q(3)\) and \(Q^{-1}(3)\). $$ Q(x)=\frac{2}{3} x-5 $$

(Graphing program required.) At low speeds an automobile engine is not at its peak efficiency; efficiency initially rises with speed and then declines at higher speeds. When efficiency is at its maximum, the consumption rate of gas (measured in gallons per hour) is at a minimum. The gas consumption rate of a particular car can be modeled by the following equation, where \(G\) is the gas consumption rate in gallons per hour and \(M\) is speed in miles per hour: \(G=0.0002 M^{2}-0.013 M+1.07\) a. Construct a graph of gas consumption rate versus speed. Estimate the minimum gas consumption rate from your graph and the speed at which it occurs. b. Using the equation for \(G,\) calculate the speed at which the gas consumption rate is at its minimum. What is the minimum gas consumption rate? c. If you travel for 2 hours at peak efficiency, how much gas will you use and how far will you go? d. If you travel at \(60 \mathrm{mph}\), what is your gas consumption rate? How long does it take to go the same distance that you calculated in part (c)? (Recall that travel distance = speed \(\times\) time traveled.) How much gas is required for the trip? e. Compare the answers for parts (c) and (d), which tell you how much gas is used for the same-length trip at two different speeds. Is gas actually saved for the trip by traveling at the speed that gives the minimum gas consumption rate? f. Using the function \(G,\) generate data for gas consumption rate measured in gallons per mile by completing the following table. Plot gallons per mile (on the vertical axis) vs. miles per hour (on the horizontal axis). At what speed is gallons per mile at a minimum? g. Add a fourth column to the data table. This time compute miles/gal \(=\mathrm{mph} /(\mathrm{gal} / \mathrm{hr}) .\) Plot miles per gallon vs. miles per hour. At what speed is miles per gallon at a maximum? This is the inverse of the preceding question; we are normally used to maximizing miles per gallon instead of minimizing gallons per mile. Does your answer make sense in terms of what you found for parts (b) and (f)?

a. Find the equation of the parabola with a vertex of (2,4) that passes through the point (1,7) . b. Construct two different quadratic functions both with a vertex at (2,-3) such that the graph of one function is concave up and the graph of the other function is concave down. c. Find two different equations of a parabola that passes through the points (-2,5) and (4,5) and that opens downward. (Hint: Find the axis of symmetry.)
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