Every point on a proportional graph tells a small story. To read it, remember what each axis measures.
Idea clave
A point $(x, y)$ means "$x$ of the first quantity goes with $y$ of the second." The point $(1, k)$ shows the unit rate, and $(0, 0)$ means none of one goes with none of the other.
Por qué funciona
The graph of $y = kx$ is made of all the pairs that fit the relationship. At $x = 1$ the equation gives $y = k$, so the unit rate always appears as the point $(1, k)$. At $x = 0$ it gives $y = 0$.
Worked examples
Ejemplo resuelto: Reading a point
Maya's pay graph (hours across, dollars up) passes through $(5, 60)$. What does that point mean?
- The first number is hours and the second is dollars.
- So in 5 hours, Maya earns \$60.
Respuesta: Working 5 hours earns \$60.
Ejemplo resuelto: Finding the unit rate point
A proportional graph of pages read (up) against minutes (across) passes through $(4, 6)$. Which point shows the unit rate?
- $k = \frac{6}{4} = 1.5$ pages per minute.
- The unit rate point is $(1, 1.5)$: in 1 minute, 1.5 pages.
Respuesta: $(1, 1.5)$
Ejemplo resuelto: Using a point to predict
The same graph passes through $(4, 6)$. How many pages are read in 10 minutes?
- $y = 1.5x$, so $y = 1.5 \times 10 = 15$.
- The point $(10, 15)$ is on the line.
Respuesta: 15 pages
Error común
Reading a point backwards. $(5, 60)$ on an hours-and-dollars graph does not mean "\$5 for 60 hours." The $x$-value always comes first. Check: does your sentence match the labels on the axes?