Chris Kirkpatrick, Barbara Alldred, Andrew Dmytriw, Shawn Godin, Angelo Lillo, David Pilmer, Susanne Trew, Noel Walker · Grade 11 · Study material generated by AI from this edition. Not affiliated with the publisher.
Chapter 4 · pp. 220–281
This chapter looks at quantities that change by a constant multiplier, such as bouncing balls, cooling drinks, doubling amounts, and money that grows or loses value. It builds the exponent laws for integer and then rational exponents, and uses them to simplify expressions. It then studies exponential functions , their graphs and transformations, and how to model growth and decay. This part covers the first ideas: the first exponent patterns, growth and decay situations, and zero and negative exponents.
Binary code uses only the digits 0 and 1. With binary digits there are possible codes. One digit gives 2 codes, two digits give 4, and three digits give 8.
Suppose you multiply the number of codes for 3 digits by the number for 2 digits. You get , which is . The exponents and add to . Dividing goes the other way: , and . So multiplying powers with the same base adds exponents, and dividing them subtracts exponents. (p. 223)
Some situations do not have constant first differences, so they are not linear. Take a bouncing ball. Each bounce reaches a fixed fraction of the previous height. For example, if each bounce is 0.6 of the last, heights of 100, 60, 36, 21.6 cm have first differences of , , . Each difference is 0.6 times the one before. This is a multiplicative pattern, and the scatter plot is a curved, decreasing, nonlinear shape. (pp. 224–225)
The heights never reach zero in the model. A fraction of a positive number is still positive, so the curve gets closer and closer to 0 without touching it. A real ball does eventually stop, but the model does not show that. (p. 225)
Domain and range must fit the situation. Bounce number 2.5 has no meaning, so only whole bounce numbers are real data. The points are discrete, and a dashed curve through them is only a guide to the shape. The range runs from 0 up to the starting height. (p. 225)
Cooling. A cup of hot cocoa left in a room cools quickly at first and then more slowly. Its temperature approaches room temperature but never drops below it. The graph has the same decreasing, flattening shape as the bounce heights. Room temperature is the horizontal asymptote. (p. 226)
Doubling. Suppose a chessboard has 1 grain of rice on the first square and the amount doubles on each square. The squares hold 1, 2, 4, 8, ... grains, and the first differences are 1, 2, 4, ..., which copy the original values. This is increasing exponential growth. The bounce data showed decay, but both have multiplicative first-difference patterns. (p. 226)
Metric units differ from one another by factors of 10, so conversions are easy. Each prefix matches a power of 10 times the metre:
Units smaller than the metre, such as the centimetre (0.01 m) and the millimetre (0.001 m), need exponents below zero. The next section shows what those mean. (p. 227)
Zero exponent. Move down the metric table. Each row divides the multiple by 10 and lowers the exponent by 1. The metre row is therefore . The quotient rule gives the same result: . Any nonzero number divided by itself is 1, so . This works for any nonzero base:
(p. 228)
Negative exponents. Continue the pattern past the metre: and . This works for any nonzero base:
For example, . (pp. 228–229)
Common mistake: a negative exponent does not make the answer negative. The exponent means "take the reciprocal." With a positive base the result stays positive, so , not . For a positive integer , is a small positive number less than 1. (p. 229)
Opposite exponents give reciprocals. Since , the product of and is . So the two powers are reciprocals. For example, and . (p. 229)
Parentheses and signs. When the base is negative, parentheses decide the sign.
(p. 229)
Fractions with negative exponents. Flip the fraction and make the exponent positive:
For example, . This works because divided by a fraction is the same as times its reciprocal. (p. 230)
For :
The product and quotient rules need the same base. All three rules still hold when exponents are zero or negative. For example, . (p. 231)
Complex numerical expressions. Simplify the numerator and denominator separately, then combine them. For :
A calculator gives the same result if you put parentheses around the whole numerator and the whole denominator. Without them, the calculator may divide at the wrong step. (p. 230)
Answer conventions. A numerical answer should be an integer, a fraction, or a decimal, so write as . An algebraic answer should use positive exponents, so write as . (p. 231)
A square has area , and its side length is the square root of . To write as a power of , let . Then . Since , the exponents must match: , so . This gives . For example, . (p. 234)
The same reasoning works for a cube. Here and , so and . That gives , the cube root. In general, . Examples: because , and because . (pp. 235–236)
The small number at the left of a radical sign is the index. It tells which root is meant. In the index is 4, so we ask which number multiplied by itself four times gives 16. The answer is 2. With no number written, the root is a square root. The index always matches the denominator of the rational exponent. (p. 236)
A useful application: the side of a cube is its volume raised to the power . A cube with volume has side m. Check: . (p. 240)
When the numerator of the exponent is not 1, split the exponent into a root and a power:
The denominator is the root, and the numerator is the ordinary power. Take :
Both orders give 9, but taking the root first keeps the numbers small. (pp. 236, 238)
For a negative or decimal exponent, use these steps:
For , write . Then
(p. 237)
Every exponent law that works for integer exponents also works for rational exponents. This includes the product, quotient, and power rules. A good strategy is to write everything as one power of the same base. Look at :
The radical became , so the exponents could be added and subtracted. (pp. 237–238)
Be careful with negative bases. An even root of a negative number has no real value, because squaring any real number gives a nonnegative result. So is not a real number. An odd root of a negative number is fine: . Rational exponents with an even denominator therefore need a nonnegative base. (p. 238)
A common mistake is thinking . Roots do not split over addition. Test it: , but . Roots do split over multiplication and division: . (pp. 238, 240)
A sphere gives a good reason to simplify. Its surface area is and its volume is . The ratio is
The cancels and the coefficient becomes . Subtracting exponents gives . So the ratio is . For it equals 1.5, the same as the long calculation but with far fewer keystrokes. Because is in the denominator, the ratio gets smaller as the radius grows. Small cells therefore have a higher ratio, which helps them stay buoyant near the surface. (pp. 241–242)
Simplify first, then substitute. Both orders give the same answer, but simplifying first needs fewer calculations. Take with and . The numerator exponent is . Subtracting the denominator exponent gives . Now , so the value is . Substituting first would mean working with large powers like twice, which is more work and more risk of error. (pp. 243–244)
Integer exponents. Apply the outer power to every factor, subtract exponents of matching bases, and finish with positive exponents. For :
(p. 243)
Rational exponents. Apply each fractional exponent to the coefficient and to every variable. Take :
(p. 244)
Several radicals. Radicals with different indices cannot be combined directly. A fifth root divided by a square root is one example. Rewrite each radical as a power, use the exponent rules, and convert back if needed. For :
(p. 245)
An exponential function has the form . The base is a constant with and . The variable sits in the exponent. Examples are and . These functions model things like population growth and a cooling liquid (p. 250). Compare with . In the variable is the base and the exponent is fixed. In the base is fixed and the variable is the exponent.
A table with equal steps in shows the type of function.
Take for , giving . The first differences are . Each one is 3 times the one before. The second differences are , which are also not constant. Equal steps in multiply by the same factor (pp. 242, 252).
You can also classify a function from its equation or graph (pp. 251–252):
For (with ), every graph of has the same key features (p. 252):
When , the function is increasing and shows growth. For , the value doubles each time goes up by 1. When , the function is decreasing and shows decay. For , the value halves each time. The graph of is the mirror image of in the y-axis (pp. 250, 252).
The size of the base affects how fast the curve changes (pp. 241, 252):
If , then for every . The graph is the horizontal line , which is not exponential. If is negative, such as , some exponents give non-real values. For example, would need the square root of a negative number. The function would then not be defined for all real (pp. 251–252).
Start with the base function . A transformed function has the form . For exponentials this is written . The parameters and change the -values. The parameters and change the -values. Changing the base also changes the shape of the curve (pp. 254, 261, 266).
The graph is stretched or compressed vertically by a factor of . If it is a stretch. If it is a compression. A negative also reflects the graph in the x-axis. For , the point on becomes . The -value is doubled and then flipped in sign (pp. 247, 261).
The graph is stretched or compressed horizontally by a factor of . If it is a compression. If it is a stretch. A negative also reflects the graph in the y-axis. For , the -values are multiplied by and reflected. So on moves to (pp. 248, 261).
In the curve moves 4 units right, and the asymptote stays at . In the curve moves 2 units left and 3 units up, so the asymptote becomes (pp. 248, 261).
To sketch a transformed exponential, follow these steps (pp. 247–248, 261):
The order matters only when two steps change the same coordinate. A vertical stretch and a horizontal shift affect different coordinates, so they can be done in either order.
A common mistake is to read as a horizontal stretch by 2 and a shift of 2 units. In fact , which is a horizontal compression by . Factoring gives . So the shift is 1 unit left, not 2. Always factor the exponent so that it looks like before reading off the transformations (p. 261).
Transformations never change the domain, which stays all real numbers. They can change the range. The range depends on the asymptote and on which side of it the curve lies. If the curve is above the asymptote, the range is . If it is below, the range is . For , the reflection puts the curve below the asymptote , so the range is . For , the curve is above , so the range is (pp. 247, 250, 261).
Two exponential equations can look different and still describe the same function. By the power-of-a-power rule, . On a graph, is compressed horizontally by a factor of . The point becomes , and becomes . Both points lie on (p. 249).
To turn words into an equation, start with the general form . Then match each phrase to a parameter: a vertical stretch or compression gives , a reflection in the y-axis gives , a horizontal shift gives , and the horizontal asymptote gives .
Example: base 2, stretched vertically by 1.5, reflected in the y-axis, asymptote . This gives , , and , so . Check with a known point. At , , which matches the given y-intercept. The domain stays all real numbers. The range is , because the curve sits above its asymptote (p. 250).
Many situations repeat the same multiplication. Write them as . Here is the initial amount, is the growth or decay factor, is the number of periods, and is the final amount (pp. 260, 276).
Why ? Adding 8% of a number to itself gives 108% of it. So each period you multiply by 1.08. For Wood Buffalo, 35 000 people in 1996 growing 8% a year gives , in thousands. After 1 year , and after 2 years (pp. 264, 265).
Telling growth from decay. Look at . In , , so this is 2% growth. In , , so this is decay at . A negative , as in , only flips the graph over. It does not make the function a decay function (pp. 270, 279).
Matching units. The time unit of the rate must be the time unit of the exponent. If light drops per metre, the exponent counts metres. If a car loses value per year, 30 months must become 2.5 years. A bacteria population that doubles every 10 h uses the exponent , because one doubling happens for every 10 hours (pp. 260, 262).
Doubling time. Use a table, a graph, or guess and check until the output is twice the start. For , is too low. is double 35, so the population doubles in about 9 years. On a graph, draw the line and find where it meets the curve (pp. 264, 265).
Half-life. Use , where is the half-life and is in the same units as . For polonium-210, days and . For 5 years, first convert to days. Then g (pp. 266, 267).
Finding the time for a given value. To find when the sample falls to 110 g, solve . You can guess and check. gives about 121 g, which is too high. gives about 110 g. So it takes about 119 days. Or graph and the line and use the intersect tool (pp. 266, 267).
Fitting data. Find from the value at . Estimate from the ratio of successive values. Then adjust by comparing the graph with the data. The frog populations 400, 480, 576 give and , so . Graph to see when the population triples. It happens at about 6 years (pp. 268, 269).
Depreciation. Losing per year means keeping , so . A V(n)=24,000(0.82)^nn=2.5V\approx$14,600$. The base is between 0 and 1, which fits decay (pp. 269, 270).
Going back in time. A negative exponent gives values before the starting time. For (thousands), a population of 8000 means . Solving gives , so about 16 years ago (p. 272).
Average growth rate. Solve for , then subtract 1. A town growing from 8400 to 12 500 in 15 years has , so about 2.7% per year. The 15th root is the rational exponent at work (p. 273).
A cooling model has the form . The start temperature is , and the asymptote is the room temperature . For , the start is and the room is . The y-intercept shows the starting temperature. The object gets closer to 19 but never reaches it (p. 272).
Zeno's paradox claims Achilles can never catch a tortoise with a 1000 m head start. Each step covers half of the remaining gap. The gap follows a halving model that shrinks toward 0, and there are infinitely many steps. But the distances add up to exactly 1000. Infinitely many steps do not need infinite distance or time, so Achilles passes the tortoise (p. 274).
The 1950 population of 2.55 billion gives . The ratio over one 5-year step is . This gives a model like , with counting 5-year periods. Graph it against the data and adjust and until it fits. It matches 1950 to 2000 well. But it cannot match the UN forecast of a peak near 2075, because a real population cannot grow without limit (p. 281).
