Slope is a rate, and it carries a unit
For a linear function, the slope between any two points is the same number: \[ m=\frac{y_2-y_1}{x_2-x_1}=\frac{\text{change in output}}{\text{change in input}} . \]
The unit of $m$ is output unit per input unit. If $h$ is centimetres and $t$ is hours, then $m$ is measured in centimetres per hour — and the interpretation sentence is forced to say “per hour”. Any answer choice missing the “per” is wrong on sight.
Mẹo đọc nhanh cả nhóm câu “interpretation”: viết đơn vị của $m$ ra lề trước khi đọc bốn phương án. Rất nhiều phương án sai chỉ vì thiếu chữ “mỗi giờ” hoặc đổi ngược thành “giờ mỗi xen-ti-mét”. Bạn loại được hai, ba phương án mà chưa cần hiểu bài toán.
A candle burns at a constant rate. Its height, in centimetres, after $t$ hours is $h(t)=30-2.5t$.
$-2.5$: the height falls by $2.5$ centimetres each hour. $30$: the height at $t=0$, i.e.\ the original height, $30$ cm. $t$-intercept: $30-2.5t=0$ gives $t=12$, so the candle burns out after $\boxed{12\ \text{hours}}$.
Three forms, and what each hands you
- Slope-intercept $y=mx+b$: slope $m$, $y$-intercept $(0,b)$.
- Point-slope $y-y_1=m(x-x_1)$: slope $m$, and the point $(x_1,y_1)$ is on the line.
- Standard $Ax+By=C$: slope $-\dfrac{A}{B}$, $\;y$-intercept $\dfrac{C}{B}$, $\;x$-intercept $\dfrac{C}{A}$.
Hoa has $120 to spend on sandwiches costing $4 each and salads costing $6 each. Her spending is modelled by $4x+6y=120$, where $x$ is sandwiches and $y$ is salads. Interpret the slope.
Slope $=-\dfrac{4}{6}=-\dfrac23$. Its unit is salads per sandwich: each extra sandwich costs her $\tfrac23$ of a salad. The intercepts are $x=30$ (all sandwiches) and $y=20$ (all salads).
Ở dạng $Ax+By=C$, hệ số góc là $-A/B$ chứ không phải $A/B$, và cũng không phải $-B/A$. Với $3x+5y=20$: hệ số góc là $-3/5$. Phương án $-5/3$ luôn có mặt, dành cho người lật ngược.
Getting the slope from a table or from two points
The table shows values of a linear function $g$.
| $x$ | $1$ | $3$ | $7$ |
|---|---|---|---|
| $g(x)$ | $11$ | $3$ | $-13$ |
Slope from the first pair: $\dfrac{3-11}{3-1}=-4$. Confirm with the second pair: $\dfrac{-13-3}{7-3}=-4$. Consistent, so $g$ really is linear. Then $g(x)=11-4(x-1)=15-4x$, and $g(0)=\boxed{15}$.
Bảng trong đề không luôn cách đều. Thấy $g$ giảm $8$ rồi giảm $16$ mà kết luận “không tuyến tính” là hỏng — phải chia cho khoảng cách $x$ tương ứng ($2$ và $4$), lúc đó cả hai đều ra $-4$.
A pool is drained at a constant rate. After $3$ minutes it holds $4{,}200$ litres; after $11$ minutes it holds $2{,}600$ litres. How much did it hold at the start, and when is it empty?
Rate: $\dfrac{2600-4200}{11-3}=\dfrac{-1600}{8}=-200$ litres per minute. Back up to $t=0$: $V(0)=4200+3(200)=4800$ litres, so $V(t)=4800-200t$. Empty when $4800-200t=0$, i.e.\ $t=\boxed{24}$ minutes.
For a linear function the average rate of change over any interval equals the slope. So a question asking for the average rate of change of a line between $x=2$ and $x=90$ is asking for $m$, and nothing else.
Intercepts are events in the story
The $y$-intercept is the value before anything happens — a start-up fee, an initial population, a full tank. The $x$-intercept is the moment the modelled quantity reaches zero — the candle burns out, the debt is paid, the tank empties.
The cost, in dollars, of producing $n$ units of a product is $C(n)=250+12n$. Give the best interpretation of $250$ and of $12$.
$250$ is the cost when $n=0$: the fixed cost incurred before a single unit is made. $12$ is dollars per unit: each extra unit adds $12 to the total cost. Note that $12$ is not the cost of the first unit — that one costs $262.
Phân biệt hai câu hỏi rất giống nhau: “chi phí của sản phẩm thứ nhất” và “chi phí tăng thêm cho mỗi sản phẩm”. Cái sau là hệ số góc; cái trước là $C(1)-C(0)$ cộng thêm phần cố định nếu đề hỏi tổng. Đề hay dùng đúng chỗ này để tạo phương án gài.
Two models at once, and moving a parameter
Account B holds $1200+85t$ dollars after $t$ months; account C holds $1830+40t$ dollars. After how many months do the two hold the same amount, and how much is that?
$1200+85t=1830+40t$ gives $45t=630$, so $t=14$ months, and the common amount is $1200+85(14)=\boxed{\$2390}$. (Check: $1830+40(14)=2390$.)
The function $f$ is defined by $f(x)=ax+b$, where $a<0$ and $b>0$. Which must be true about the $x$-intercept of the graph of $f$?
The $x$-intercept is $x=-\dfrac{b}{a}$. With $b>0$ and $a<0$, the quotient $b/a$ is negative, so $-b/a$ is positive. The $x$-intercept is positive, whatever the actual values.
Steepness is $|m|$; “greater slope” is $m$ itself. A line of slope $-5$ is steeper than a line of slope $2$, yet $2$ is the greater slope. Both statements are true at the same time, and the test writes answer choices that mix them.
The graph of $y=2x-6$ is changed by increasing the constant term from $-6$ to $-1$, leaving the coefficient of $x$ alone. What happens to the $x$-intercept?
Before: $2x-6=0$ gives $x=3$. After: $2x-1=0$ gives $x=\tfrac12$. The $x$-intercept moves left, from $3$ to $\tfrac12$. In general the $x$-intercept is $-b/m$, so raising $b$ while $m>0$ pushes the intercept toward $-\infty$; the whole graph slides up, which drags the crossing point backwards. Answer: it decreases.
Đề rất hay hỏi “nếu tăng $b$ thì hoành độ gốc đổi thế nào”. Đừng suy luận bằng lời — viết $-b/m$ ra rồi thay hai con số cụ thể. Ba mươi giây, và không phụ thuộc vào việc bạn hình dung đồ thị đúng hay sai.
A shop's monthly profit, in dollars, is $P(x)=18x-2700$, where $x$ is the number of items sold. How many items must be sold for the shop to break even, and what does $-2700$ represent?
Break-even is the $x$-intercept: $18x=2700$ gives $x=\boxed{150}$ items. The constant $-2700$ is the profit when nothing is sold — a loss of $2,700, i.e.\ the monthly fixed costs.
In a story, the input is whatever the sentence says “per” about. “$18 per item” makes items the input, so $x$ counts items and the $18 is the slope — never the other way round, however the letters are named.