Linear Exponential Patterns Quiz: Distinguishing Linear and Exponential Patterns

Exponential functions y = a times b^t have a constant percent change of (b-1) times 100 percent per step. Here b=1.07 gives a constant 7% increase per unit time. Option A is linear with constant additive change of -7 per step, not constant percent change. Option C is quadratic with varying rate of change. Option D is logarithmic with decreasing rate of change.

In y = mx + b, the coefficient m is the slope, which equals the constant amount added per unit increase in x. Here m = 2.5, so each time x increases by 1, y increases by 2.5. Option A gives 7, which is the y-intercept not the slope. Option B gives 1.5, which has no relationship to the function. Option C gives 3.5, also unrelated to the given coefficients.

The answer is True. In the linear model y = mt + b, a negative value of m means the quantity decreases by a fixed amount of

Consecutive ratios: 180/240=0.75, 135/180=0.75, 101.25/135=0.75. The constant ratio is 0.75 per step, confirming exponential decay. Each term is 75% of the previous, equivalent to a 25% decrease per step. Option A gives 0.50, producing 240, 120, 60. Option B gives 0.65, producing 240, 156. Option D gives 0.85, producing 240, 204. Only 0.75 generates the given sequence.

The answer is True. A constant ratio r between consecutive terms means each term is r times the previous term. Starting from y_0, this gives y_n = y_0 times r^n, which is exactly the definition of an exponential sequence. This is the ratio test for identifying exponential patterns, just as constant differences identify linear patterns.

The equation has the form y = mx + b with slope m = -0.4 and y-intercept 7. This is a linear function with a constant additive change of -0.4 per unit increase in x. Option B is incorrect because x is not an exponent. Option C would require an x² term. Option D is incorrect because the equation perfectly matches the standard linear form.

Consecutive differences: 9-5=4, 13-9=4, 17-13=4. The difference is constant at 4 per step, which is the defining feature of a linear pattern. Exponential patterns have a constant ratio, not a constant difference. Option C would apply if neither difference nor ratio were constant. Option D is incorrect because the pattern is clearly identifiable.

At n=0, 3^0=1, so y(0) = 5 times 1 = 5. Option A gives 1, which is just 3^0 without multiplying by the coefficient 5. Option B gives 3, which is the base alone. Option D gives 15 = 5 times 3, which would be the value at n=1 not n=0. The initial value of any exponential function y=a times b^n is always a when n=0.

The answer is False. The function y = a times b^t multiplies by the constant factor b each step. Here b=1.02 means the quantity is multiplied by 1.02 each step, which is a 2% multiplicative increase. Adding 2 units would be a linear model such as y = 500 + 2t. Exponential and linear models differ fundamentally in whether they multiply or add a constant each step.

Ratios: 160/200=0.8, 128/160=0.8, 102.4/128=0.8. The constant ratio of 0.8 per step confirms an exponential model. Since 0.8 is less than 1, the values are decreasing, confirming exponential decay with decay factor 0.8, equivalent to a 20% decrease per step. The differences are -40, -32, -25.6 which are not constant, ruling out linear decay.

Differences: 90-100=-10, 80-90=-10, 70-80=-10. The constant difference of -10 per step confirms a linear model. Since the values are decreasing, it is linear decay with slope -10 per unit time. Exponential decay would show a constant ratio, but here the ratio 90/100=0.9, 80/90=0.889 is not constant, ruling out option D.

Linear functions have the form y = mx + b with a constant slope. Option A has slope 4, confirming it is linear. Option D has slope -5, also linear. Option B is exponential because x appears as an exponent. Option C is quadratic because of the x² term — the slope changes with x, making it nonlinear.

Ratios: 6/2=3, 18/6=3, 54/18=3. The constant ratio is 3 per step, confirming exponential growth with base 3. Option A gives 2, which would produce the sequence 2, 4, 8, 16. Option C gives 4, producing 2, 8, 32. Option D gives 6, producing 2, 12, 72. Only 3 generates the given sequence.

Differences: 10-7=3, 13-10=3, 16-13=3. The constant difference is 3 per step, confirming this is a linear sequence. Option A gives 1, option B gives 2, and option D gives 4, none of which match the actual differences between consecutive terms.

The answer is True. A linear model has the form y = mx + b, where m is the constant amount added per unit of time. Equal additive change per interval is the defining characteristic of linear growth. Exponential growth multiplies by a constant factor each interval rather than adding a constant amount.

Consecutive ratios: 6/3=2, 12/6=2, 24/12=2. The ratio is constant at 2 per step, which is the defining feature of an exponential pattern. A linear pattern would require a constant difference, but the differences here are 3, 6, 12 which are not constant. Option C would apply if neither test passed.

Option A has ratios 27/81=1/3, 9/27=1/3, 3/9=1/3 — constant ratio of 1/3, confirming exponential decay. Option D has ratios 3/1=3, 9/3=3, 27/9=3 — constant ratio of 3, confirming exponential growth. Option B has constant difference of 5, making it linear. Option C has differences 1, 2, 3 and ratios 1.5, 1.667 — neither constant, so neither linear nor exponential.

A constant 8% growth per day means multiplying by 1.08 each day: y = 50 times (1.08)^t. Option A adds a constant 0.08 units per day, which is linear not exponential and also an extremely small amount. Option C subtracts 0.08 per day, which is linear decay. Option D uses 0.92 as the factor, which represents 8% decay per day, not growth.

Linear functions add the same amount per step, confirming A. Exponential functions multiply by the same factor per step, confirming B. When the base b is between 0 and 1, repeated multiplication shrinks the quantity confirming exponential decay, confirming C. Option D is false — linear functions have constant additive change, so their percent change varies depending on the current level. As values grow or shrink, the same additive change represents a different percentage.

Differences: 8-5=3, 12-8=4, 18-12=6. These are not constant, ruling out linear. Ratios: 8/5=1.6, 12/8=1.5, 18/12=1.5. These are not constant, ruling out exponential. Since neither constant difference nor constant ratio exists, neither a linear nor an exponential model fits exactly. Option D is incorrect because the pattern is analyzable — it simply fits neither category.