Chapter 5 — Functions, Graphs and Exponential Growth

Answer: D

Explanation

Replace every $x$ with $-2$, using brackets so the signs survive:

$$f(-2) = 3(-2)^2 - 4(-2) + 1$$

Take it one piece at a time:

$$(-2)^2 = 4, \text{ so } 3(4) = 12$$ $$-4(-2) = +8$$

$$f(-2) = 12 + 8 + 1 = 21$$

Two sign points do all the damage here. $(-2)^2$ is $+4$, because a negative times a negative is positive. And $-4 \times -2$ is $+8$, for the same reason. A negative input often produces a larger answer than you expect.

Why each wrong option is wrong:

• A, $-19$ — both sign slips at once: $-12 - 8 + 1$.
• B, $5$ — squared correctly but lost the minus in the middle term, computing $12 - 8 + 1$. Subtracting a negative adds.
• C, $-3$ — squared $-2$ as $-4$, giving $-12 + 8 + 1$. The bracket matters: $(-2)^2 = 4$, while $-2^2 = -4$.

Takeaway: Substitute inside brackets, then simplify. A negative squared is positive, and subtracting a negative adds.

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Answer: A

Explanation

Work from the inside out. The inner function is $g$:

$$g(3) = 3^2 = 9$$

Now put that 9 into $f$:

$$f(9) = 2(9) + 1 = 19$$

Why each wrong option is wrong:

• C, $49$ — did $f$ first: $f(3) = 7$, then $g(7) = 49$. That is $g(f(3))$, a different question — and it is offered every single time.
• B, $13$ — found $2 \times 9$ and forgot the $+1$.
• D, $36$ — multiplied $f(3)$ and $g(3)$ together. Composition is not multiplication.

Takeaway: $f(g(x))$ means $g$ first. Work outward from the innermost bracket, exactly as with ordinary arithmetic.

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Answer: 7

Explanation

$f(a) = 27$ means the output is 27, so set the rule equal to 27 and solve for the input:

$$5a - 8 = 27 \quad\Longrightarrow\quad 5a = 35 \quad\Longrightarrow\quad a = 7$$

Check: $f(7) = 35 - 8 = 27$ ✓

This is the reverse of Q1. There you were given the input and asked for the output; here you are given the output and asked for the input. Notice which one the notation is telling you: the number inside the brackets is the input, the number the function equals is the output.

This is a grid-in. Enter 7.

Takeaway: $f(a) = k$ means "set the rule equal to $k$ and solve". The number inside the brackets is always the input.

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Section 2 — Reading and Transforming Graphs

Changing a function's rule moves its graph in predictable ways. The one that catches everybody is the horizontal shift, because it goes the opposite way to what the sign suggests.

The rule becomes	The graph
$f(x) + k$	moves up $k$
$f(x) - k$	moves down $k$
$f(x + k)$	moves left $k$ — opposite to the sign
$f(x - k)$	moves right $k$ — opposite to the sign
$-f(x)$	flips vertically, over the $x$-axis
$f(-x)$	flips horizontally, over the $y$-axis
$af(x)$, $a>1$	stretches vertically — taller

Why does $f(x - 3)$ move the graph right? Because to get the same output you now need an input 3 larger. Whatever $x$ used to do the job, $x + 3$ does it now. The change is inside the bracket, so it affects the input, and inputs move opposite to the sign.

The reliable way to settle any transformation question: track one point. Pick a point you know is on the original graph and work out where it lands.

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Answer: B

Explanation

The change is inside the bracket, so it is a horizontal shift — and inside changes go the opposite way to the sign. $f(x - 3)$ moves the graph right 3.

Track the point. Let $g(x) = f(x - 3)$. We know $f(2) = 5$, so we want the input that makes the bracket equal 2:

$$x - 3 = 2 \quad\Longrightarrow\quad x = 5$$

Then $g(5) = f(5 - 3) = f(2) = 5$. So $g$ passes through $(5, 5)$.

The $y$-value does not change at all — a horizontal shift moves points sideways.

Why each wrong option is wrong:

• A, $\left( -1, \ 5\right)$ — moved left, following the minus sign. Inside the bracket, the sign lies.
• C, $\left( 2, \ 8\right)$ — moved up 3, which is what $f(x) + 3$ would do. Outside the bracket affects the output.
• D, $\left( 2, \ 2\right)$ — moved down 3.

Takeaway: Inside the bracket → horizontal → opposite to the sign. Outside → vertical → as the sign says. Track a single known point and the question answers itself.

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Answer: A

Explanation

The $+6$ is outside the bracket, so it changes the output: every point moves up 6. The $x$-coordinate is untouched.

$$(1, -4) \longrightarrow (1, -4 + 6) = (1, 2)$$

The minimum stays a minimum — shifting a graph up does not change its shape, only its position.

Why each wrong option is wrong:

• B, $\left( 7, \ -4\right)$ and D, $\left( -5, \ -4\right)$ — moved sideways. A change outside the bracket never moves a graph sideways.
• C, $\left( 1, \ -10\right)$ — subtracted 6 instead of adding. $-4 + 6 = 2$, not $-10$.

Takeaway: Outside the bracket moves the graph vertically, in the direction the sign says. Add to $y$, leave $x$ alone.

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Answer: D

Explanation

The minus is outside the function, so it acts on the output. Every $y$-value changes sign; the $x$-values stay where they are.

$$(-3, 7) \longrightarrow (-3, -7)$$

This is a reflection over the $x$-axis: the graph is flipped upside down.

Compare with $y = f(-x)$, where the minus is applied to the input. That reflects over the $y$-axis and would send $(-3, 7)$ to $(3, 7)$ — which is option A, and it is offered precisely because the two reflections are easy to swap.

	Flips	Point $(-3, 7)$ becomes
$-f(x)$	over the $x$-axis	$(-3, -7)$
$f(-x)$	over the $y$-axis	$(3, 7)$

Why each wrong option is wrong:

• A, $\left( 3, \ 7\right)$ — that is $f(-x)$, the other reflection.
• B, $\left( 3, \ -7\right)$ — flipped both coordinates, which is neither transformation.
• C, $\left( 7, \ -3\right)$ — swapped $x$ and $y$, which is what an inverse does.

Takeaway: The minus outside negates the output; the minus inside negates the input. Ask which one the minus is attached to before deciding which axis it flips over.

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Section 3 — Exponential Growth and Decay

An exponential model multiplies by a fixed factor each period, where a linear model adds a fixed amount.

$$y = a \cdot b^{\,x}$$

• $a$ is the starting value, the amount at $x = 0$.
• $b$ is the growth factor — what you multiply by each period.

Turning a percentage into a factor is where the marks are:

The situation	$b$ is
grows by 8%	$1.08$
grows by 50%	$1.5$
falls by 8%	$0.92$
falls by 30%	$0.70$
doubles	$2$
halves	$0.5$

The rule: increase means $1 + r$, decrease means $1 - r$. A factor of $1.08$ is not "8%" — it is "100% of what you had, plus 8% more".

Growth means $b > 1$. Decay means $0 < b < 1$. If a model has $b = 0.85$ the quantity is shrinking by 15% each period, not growing by 85%.

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Answer: B

Explanation

Growth by a percentage each period is exponential, so the shape is $y = a \cdot b^h$.

The starting value is $a = 400$. Growing by 15% means each hour the population becomes 100% + 15% = 115% of what it was, so

$$b = 1.15$$

$$P = 400(1.15)^h$$

Check the first hour: $400 \times 1.15 = 460$, which is $400 + 60$, and 60 is 15% of 400 ✓ After a second hour: $460 \times 1.15 = 529$. Note the second hour added 69, not 60 — that is the difference between multiplying and adding, and it is what makes the model exponential.

Why each wrong option is wrong:

• A, $400 \left(\frac{3}{20}\right)^{h}$ — used $0.15$ as the factor. That would multiply the colony by $0.15$ each hour, wiping out 85% of it hourly.
• C, $\frac{3 h}{20} + 400$ — a linear model, adding $0.15$ each hour. It also adds a fraction of a bacterium per hour, which should look wrong.
• D, $400 \left(\frac{17}{20}\right)^{h}$ — used $1 - 0.15$, which is the decay factor. The colony is growing.

Takeaway: Growth by $r$ means multiply by $1 + r$ each period. Check by computing the first period by hand — it should be the starting value plus exactly $r$ of it.

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Answer: C

Explanation

The factor is $0.88$, which is less than 1, so this is decay. But $0.88$ is what remains, not what is lost:

$$1 - 0.88 = 0.12$$

So the car loses 12% of its value each year, keeping 88%.

Check with real numbers. After one year:

$$24000 \times 0.88 = 21120$$

The drop is $24000 - 21120 = 2880$, and $\frac{2880}{24000} = 0.12$ ✓

Why each wrong option is wrong:

• A, The car loses 88% of its value each year — read the factor as the loss. Losing 88% a year would leave the car worth $2880 after one year, not $21120.
• B, The car loses $88 each year — a fixed dollar loss would be a linear model. The actual drop is $2880 in the first year and less in the second, because 12% of a smaller number is smaller.
• D, The car gains 88% of its value each year — a factor below 1 always shrinks.

Takeaway: The factor is what survives. Subtract it from 1 to get the percentage lost. And a fixed percentage change is exponential; a fixed amount is linear.

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Answer: A

Explanation

Test both possibilities against the table.

Is it linear? Look at the differences: $15 - 5 = 10$, $45 - 15 = 30$, $135 - 45 = 90$. Not constant, so not linear.

Is it exponential? Look at the ratios: $\frac{15}{5} = 3$, $\frac{45}{15} = 3$, $\frac{135}{45} = 3$. Constant, so yes — it multiplies by 3 each step.

The model is $y = 5 \cdot 3^x$, and the next value is

$$135 \times 3 = 405$$

Why each wrong option is wrong:

• B and D — call it linear, which the differences rule out immediately.
• C, Exponential, and $y = 270$ — recognised the exponential pattern and then doubled instead of tripling.

Takeaway: Check the differences first: constant means linear. If they are not, check the ratios: constant means exponential. Two divisions settle which kind of model you are looking at.

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Answer: 12800

Explanation

Do not multiply by 2 twenty times. Work out how many doubling periods fit into 20 years:

$$\frac{20}{5} = 4 \text{ periods}$$

Each period doubles, so four periods multiply by $2^4 = 16$:

$$800 \times 16 = 12800$$

Written as a model, that is $P = 800 \cdot 2^{t/5}$, and the $t/5$ is exactly this "how many periods" count.

Check by stepping: $800 \to 1600$ (5 years) $\to 3200$ (10) $\to 6400$ (15) $\to 12800$ (20) ✓

This is a grid-in. Enter 12800. Five characters, which is exactly the grid's limit for a positive answer.

Takeaway: When the period is not 1, divide the time by the period length to count the periods. The exponent is a count of periods, never the raw time.

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Section 4 — Mixed Practice

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Answer: B

Explanation

$$g(-4) = \frac{-4 + 6}{2} = \frac{2}{2} = 1$$

A got the numerator's sign wrong. C ignored the minus on the input. D divided the $-4$ by 2 before adding the 6.

Takeaway: Do everything on the top before dividing. The fraction bar groups the whole numerator.

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Answer: 3

Explanation

Set the rule to zero:

$$2^x - 8 = 0 \quad\Longrightarrow\quad 2^x = 8$$

Now ask: 2 to what power gives 8? Since $2^3 = 8$, the answer is $x = 3$.

This is a grid-in. Enter 3.

Takeaway: A "zero" of a function is an input making the output 0. With a power, rewrite both sides with the same base and compare exponents.

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Answer: C

Explanation

The 2 multiplies the output, so every $y$-value doubles and the $x$-values stay put:

$$(4, 3) \longrightarrow (4, 6)$$

A and B changed $x$, which only happens when the change is inside the bracket. D halved it.

Takeaway: A number multiplying $f(x)$ from outside stretches the graph vertically. Outside acts on $y$; inside acts on $x$.

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Answer: D

Explanation

The factor is $1.05$. Subtract 1 to get the rate:

$$1.05 - 1 = 0.05 = 5\%$$

So the investment grows by 5% each year, starting from $2000.

Check: year one gives $2000 \times 1.05 = 2100$, a gain of $100 — and $100 is 5% of $2000 ✓ Year two gives $2100 \times 1.05 = 2205$, a gain of $105. The gain grows, which is what compounding means.

A is that first-year figure misread as fixed — and notice option A's $105 is the second year's gain, not the first. B read the factor as a percentage directly. C read the factor as the starting amount, which is the 2000.

Takeaway: In $a \cdot b^n$, $a$ is the start and $b - 1$ is the rate. An exponential gain is never a fixed amount — that is precisely what separates it from a linear one.

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Answer: C

Explanation

Check the differences first: $4 - 7 = -3$, $1 - 4 = -3$, $-2 - 1 = -3$. They are constant, so the function is linear and the rate is $-3$.

The starting value is $f(0) = 7$, which is the constant term. So

$$f(x) = -3x + 7$$

Test every row: $f(2) = -6 + 7 = 1$ ✓

Why each wrong option is wrong:

• A, $y = 7 x - 3$ — swapped the two numbers, using 7 as the rate and $-3$ as the start. At $x = 0$ it gives $-3$, not 7.
• B, $y = 4 - 3 x$ — took $f(1) = 4$ as the starting value. The start is at $x = 0$.
• D, $y = 3 x + 7$ — right starting value, wrong sign on the rate. The outputs are falling, so the rate is negative.

Takeaway: With a table, take the differences. Constant differences mean linear; the difference is the rate, and $f(0)$ is the constant term.

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Answer: A

Explanation

An $x$-intercept is a point where the graph meets the $x$-axis, and on that axis $y = 0$. So "crosses at $x = -2$" means the input $-2$ gives the output 0:

$$f(-2) = 0 \qquad f(5) = 0$$

Why each wrong option is wrong:

• B, $f(0) = -2$ and $f(0) = 5$ — swaps input and output, and is impossible besides: a function gives each input exactly one output, so $f(0)$ cannot be both $-2$ and 5.
• C, The graph passes through $(0, -2)$ and $(0, 5)$ — describes $y$-intercepts, and for the same reason there can only ever be one of those.
• D, $f(-2) = 5$ — invents a relationship between the two intercepts.

Takeaway: $x$-intercept means output zero: $f(a) = 0$. The number given is the input. And a function has at most one $y$-intercept, ever.

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Answer: D

Explanation

The test is simple: does the quantity go up by the same amount each step, or by the same percentage?

• Same amount added → linear
• Same percentage, or same multiplying factor → exponential

A, B and C all add a fixed amount per unit: $2,000 a year, 5 litres a minute, $2 a kilometre. All linear.

D rises by 3% of whatever it currently is, so the amount added grows as the population grows. That is exponential.

Put numbers on it. A population of 40,000 rising 3% a year gains 1,200 in the first year and 1,236 in the second — a bigger gain each time, from the same percentage.

Takeaway: "Per year", "a minute", "a kilometre" attached to a fixed amount means linear. A percentage each period means exponential.

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Answer: B

Explanation

Four days of tripling multiplies by $3^4 = 81$. So

$$81 \times \text{start} = 810 \quad\Longrightarrow\quad \text{start} = 10$$

Check forwards: $10 \to 30 \to 90 \to 270 \to 810$ ✓ Four steps.

Why each wrong option is wrong:

• A, $270$ — divided by 3 once, which undoes one day, not four. Notice 270 is the count after three days, so it is a real number from the sequence and the wrong one.
• C, $\frac{135}{2}$ — used $3 \times 4 = 12$ rather than $3^4$. Repeated multiplication is a power, not a product.
• D, $30$ — divided by 27, undoing three days.

Takeaway: Working backwards through $n$ periods means dividing by $b^n$, not by $b$ or by $bn$. Then step forwards to check.

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Answer: C

Explanation

Inside first:

$$f(3) = 2(3) - 1 = 5$$

Then feed 5 back in:

$$f(5) = 2(5) - 1 = 9$$

A stopped after one application. D squared instead of applying $f$ again.

Takeaway: $f(f(x))$ means apply $f$, then apply it again to the result — not square it.

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Answer: A

Explanation

Two separate changes, and they behave differently.

Right 4 is horizontal, so it goes inside the bracket, and inside changes run opposite to the sign. Right means $-4$: $f(x - 4)$.

Down 2 is vertical, so it goes outside, and outside changes follow the sign. Down means $-2$.

$$y = f(x - 4) - 2$$

Test it on a graph you know. Take $f(x) = x^2$, whose lowest point is $(0, 0)$. The new graph is $(x-4)^2 - 2$, whose lowest point is at $x = 4$ with value $-2$ — right 4 and down 2, exactly as asked ✓

Why each wrong option is wrong:

• B, $y = f(x + 4) - 2$ — $f(x + 4)$ moves the graph left.
• C, $y = f(x - 4) + 2$ — moves up rather than down.
• D, $y = f(x - 2) - 4$ — swapped the two numbers, moving right 2 and down 4.

Takeaway: Inside the bracket, horizontal and opposite to the sign. Outside, vertical and as the sign reads. Test with $y = x^2$ if you are unsure — its lowest point makes the move obvious.

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Answer: D

Explanation

Work out a few years of each.

Year	A (adds $100)	B (grows 8%)
0	1000	1000
1	1100	1080
3	1300	≈1260
10	2000	≈2159
30	4000	≈10,063

A is ahead early, because $100 is more than 8% of $1,000. But B's yearly gain grows — 8% of a bigger number each time — while A's stays at $100 forever. So B eventually passes A and then pulls away.

This is the general rule, and it is worth knowing: exponential growth always overtakes linear growth eventually, no matter how small the percentage or how large the fixed amount. It just may take a while.

Takeaway: Linear gains are fixed; exponential gains grow. Whenever a question compares the two over a long enough period, the exponential one wins in the end.

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Answer: B

Explanation

A fraction is undefined only when its denominator is zero. Dividing by zero has no meaning; a zero on top is perfectly fine.

$$x - 5 = 0 \quad\Longrightarrow\quad x = 5$$

A used the numerator. At $x = -3$ the function is defined and equals 0 — a real value, not an undefined one.

Takeaway: Undefined means the bottom is zero. Set the denominator to zero and solve; ignore the top entirely.

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Answer: 100

Explanation

Exponential means multiplying by the same factor each step. Two steps take 200 down to 50, so

$$200 \times b^2 = 50 \quad\Longrightarrow\quad b^2 = \tfrac14 \quad\Longrightarrow\quad b = \tfrac12$$

One step from 200 gives

$$200 \times \tfrac12 = 100$$

Check: $200 \to 100 \to 50$ ✓

There is a neat way to see this. In an exponential sequence, each term is the geometric mean of its neighbours — the square root of their product:

$$\sqrt{200 \times 50} = \sqrt{10\,000} = 100$$

Compare that with a linear sequence, where the middle term would be the ordinary average, $\frac{200 + 50}{2} = 125$. The two are different, and which one applies depends entirely on which kind of model you were given.

This is a grid-in. Enter 100.

Takeaway: In an exponential model the middle of three evenly spaced values is their geometric mean; in a linear one it is their ordinary average. Check which model the question named.

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Answer: 3

Explanation

"$f(x) = g(x)$" means set the two rules equal:

$$4x - 1 = x + 8 \quad\Longrightarrow\quad 3x = 9 \quad\Longrightarrow\quad x = 3$$

Check: $f(3) = 11$ and $g(3) = 11$ ✓ The two lines cross at $(3, 11)$.

This is the same idea as a system of equations, and the same idea as "where do the graphs meet" — three ways of describing one question. The built-in graphing calculator answers it in seconds: graph both and read the intersection.

This is a grid-in. Enter 3.

Takeaway: "Where are two functions equal" means set the rules equal to each other. It is also exactly what an intersection point is, so it can be graphed instead.

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