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This article was written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With more than eight years of teaching experience, Mario dabbles in statistical biology, optimization, mathematical modeling of genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught in high school and university.

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What Is Growth Factor In Math

Scale ratio, or scale ratio, is the ratio of two corresponding sides of the same length. Similar figures have the same shape but different sizes. Scale is used to solve geometric problems.

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You can use a scale to find the length of the missing sides of the figure. Instead, you can use the side lengths of two identical numbers to calculate the scale factor. These problems involve multiplication or require the simplification of fractions.

To find the scale, first find the length of the corresponding side in each image. If you are going from a small to a large number, enter the length in the equation scale factor = greater length than less length. If you are reducing from a larger number to a smaller one, use the equation ratio = smaller length over larger length. Connect the length and simplify the fraction to get the scale. If you want to learn how to find the scale factor in chemistry, read on! Calculate the compound interest. Step 1 – Know the goals. Step 2 – Fully understand the goals. Level 3 – Use simple problem solving objectives. Step 4 – Use goals to solve key problems. Level 5 – Prepare and apply goals to different problems and difficulties.

4 Exponential function exponential function – is a function of the form f(x) = abx, where u ≠ 0, b is an integral number and b  1. The exponential function of the parent is f(x) = bx, where the base b is the variable and the -exponent x is an independent variable.

What Is A Growth Factor?

A function of the form f(x) = abx, with > 0 and b > 1, is an exponential function, increasing as x increases. y x (0, 1) Range: (–, ) Range: (0, ) 4 increase b > 1

Growth factor – the b value of exponential growth y = abx, and b > 1. The quantity that shows exponential growth increases by a constant percentage (or rate) in each period. The percentage increase r, written as a decimal, is the growth rate or growth rate. Apparent growth b = 1 + r.

I took the graph of linear and quadratic functions. Now we will graph exponential functions. They are different from any other types of functions we have studied because the independent variable is in the exponent. Let’s look at the graph of this function by plotting some points. x 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 BASE Remember what a negative exponent means: /2/4/8

8 Asymptote Notice that as the number of x decreases, the graph of the function gets closer and closer to the x-line. The function will never reach the x line because the value of 2x cannot be zero. In this case, the x-axis is the asymptote. Asymptote – is the line that the graph of the function approaches as the value of x becomes very large or very small.

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10 x EXAMPLE 1 Graph y = b for b > 1 Graph y = x 2 SOLUTION STEP 1 Make a list of values. STEP 2 Plot the points in the table. STEP 3 Draw, from left to right, a smooth curve starting just above the x-axis, passing through the drawn points, and going up to the right.

SOLUTION Plot (1, 2) and, from left to right, draw a curve that starts just above the x-axis, passes through two points, and moves up to the right. 0, 12 a.

SOLUTION Plot (0, -1) and Then, from left to right, draw a curve that starts just below the x-axis, passes through two points, and goes down to the right. b. 1, – 5 2

You can do the same conversions for exponential functions that you did for quadratic functions. To graph a function of the form 𝑦=𝑎 𝑏 𝑥−ℎ +𝑘, first draw the graph of 𝑦=𝑎 𝑏 𝑥 . Then translate the graph horizontally by h units and vertically by k units.

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Transformation Equation Definition Vertical transformation g(x) = abx + k Shift the graph of f (x) = bx up k units if k > 0. Shift the graph of f (x) = bx down k units if k < 0. horizontal translation g(x) = bx-h Moves the graph of f (x) = bx h units to the left if h 0.

F(x) = abx-h+k Horizontal transformation in units of h (opposite side of sign) Vertical transformation in units of k (same side of sign) y = k equation of horizontal asymptote

Graph y = ab k for b > 1 x–h EXAMPLE 3 Graph y = – 3. State the area and range. x – 1 SOLUTION Begin by drawing the graph of y = , passing through (0, 4) and (1, 8). Then shift the graph to the right by 1 unit and down by 3 units to get the graph of y = – 3. The asymptote of the graph is the line y = -3. The domain is all real numbers and the range is y > -3. x – 1

Graph the function. Define range and range. 1. y = 4 x ANSWER Plot the points (0, 1) and (1, 4) •

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Graph the function. Define range and range. 2. y = 1 2 x ANSWER • Plot the points (0, ½) and (1, 3/2)

Graph the function. Define range and range. f (x) = x + 1 Plot the points of the parent function f(x) = 3x: (0, 1) & (1, 3). Translate one unit left (subtract 1 for each value of x) and 2 units up (add 2 for each value of k) and plot the points (-1, 3) and (0, 5). The new asymptote is y = 2. ANSWER •

MULTIPLE GROWTH METHODS An amount grows faster if it increases by the same percentage each time. EXTRA GROWTH SHOW a is the initial amount. t is the time interval. f(x) = a (1 + r)t (1 + r) is the growth factor, r is the growth rate. The growth percentage is 100r.

21 EXAMPLE 4 Multi-step troubleshooting In 1996, there were 2573 computer viruses and other computer security incidents. Over the next 7 years, the number of incidents increased by approximately 92% per year. Computers • Write a growth model that gives the number of events in t years after how many events in 2003?

Exponential Growth And Decay

EXAMPLE 4 Solve a Multistep Problem • Graph the model. • Use the chart to estimate the year in which approximately 125,000 computer security incidents occur. SOLUTION STEP 1 The initial amount is = 2573 and the percentage increase is r = So the exponential growth pattern: n = a (1 + r) t Write the exponential growth pattern. = 2573( ) t Substitute 2573 for a and 0.92 for r = 2573(1.92) t Simplify.

EXAMPLE 4 Solving a Multistage Problem Using this model, you can estimate the number of events in 2003 (t = 7) to be n = 2573 (1.92) .485. 7 STEP 2 The graph passes through the points (0, 2573) and (1, ). Make a few more points. Then draw a smooth curve through the points.

24 EXAMPLE 4 Multistage Problem Solving STEP 3 Using the graph, you can estimate that the number of incidents was about 125,000 in the year 2002 (t 6).

25 Your turn: eg. 4 4. What if? In Example 4, estimate a year in which approximately 250,000 computer security incidents occur. 2003 ANSWER

Dependent And Independent Variables

Your Chance: Example 4 5. For the exponential growth model y = 527(1.39) , identify the initial value, the growth rate, and the percent increase. x Initial Amount: 527 Growth Factor 1.39 Percentage Growth 39% ANSWER

27 Compound Interest Simple Interest: Money paid or earned by using money over a period of time.

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