A demand curve can be written two ways, and textbooks, exams and later courses use both. Switching between them is the skill from Solving for a Variable: solve for the other letter. You met the first form in The Law of Demand.
Key idea
$$Q_d = a - bP \quad\Longleftrightarrow\quad P = \frac{a}{b} - \frac{1}{b}\,Q.$$ The quantity intercept is $a$ (the quantity demanded at a price of zero). The price intercept is $a/b$ (the price at which no one buys, sometimes called the choke price). With price on the vertical axis, the slope of the graph is $-\dfrac{1}{b}$, not $-b$.
Why it works
Start from $Q = a - bP$. Add $bP$ and subtract $Q$: $bP = a - Q$. Divide by $b$: $P = \dfrac{a}{b} - \dfrac{Q}{b}$. That is the line as the graph draws it, with $P$ up and $Q$ across, so its slope is the coefficient of $Q$, $-\tfrac{1}{b}$. The number $b$ says how many units of quantity are lost per \$1 of price; its reciprocal says how many dollars of price per unit of quantity.
Worked example: From quantity form to price form
Demand is $Q_d = 20 - 2P$. Find the price intercept, the quantity intercept, the slope of the graph and the inverse form.
- Quantity intercept: set $P = 0$: $Q = 20$.
- Price intercept: set $Q = 0$: $0 = 20 - 2P$, so $P = 10$.
- Solve for $P$: $2P = 20 - Q$, so $P = 10 - \tfrac{1}{2}Q$.
- Slope with $P$ on the vertical axis: $\dfrac{0 - 10}{20 - 0} = -\dfrac{1}{2}$.
Answer: Intercepts 20 units and \$10; slope $-\tfrac{1}{2}$; $P = 10 - \tfrac{1}{2}Q$.
Worked example: From price form to quantity form
Inverse demand is $P = 12 - 0.5Q$. Write it as $Q_d$ in terms of $P$, and find the quantity demanded at \$8.
- Add $0.5Q$ and subtract $P$: $0.5Q = 12 - P$.
- Multiply by 2: $Q_d = 24 - 2P$.
- At \$8: $Q_d = 24 - 16 = 8$. Check in the price form: $12 - 0.5(8) = 8$. ✓
Answer: $Q_d = 24 - 2P$; 8 units at \$8.
Common mistake
Taking the $-2$ in $Q = 20 - 2P$ as the slope of the graph, or swapping the intercepts. If you learned $y = mx + b$, it is natural to read $Q$ as $y$, but on these graphs price is "$y$." Check: plot both intercepts, $(20, 0)$ and $(0, 10)$, and compute rise over run: $-10/20 = -\tfrac{1}{2}$.