Mathematics RU

Reading graphs

Read a function's value from its graph, the slope and the equation of a line through two points, then the formula of a familiar function that was shifted, stretched or flipped.

How to solve it

A graph is all the points $(x,\ f(x))$. So a value is read along a vertical line: go from $x$ up or down to the curve and read the height. And the equation of a line is made of two numbers: the slope $k$ and where it crosses the $y$-axis.

Step by step

The value $f(a)$: draw the vertical line $x = a$ to the graph and read the $y$-coordinate of the crossing.
The slope of the line through $A$ and $B$: rise over run, $k = \frac{y_B - y_A}{x_B - x_A}$.
The intercept $b$: put the coordinates of one point into $y = kx + b$ and solve for $b$ (the height where the line crosses the $y$-axis).
A shifted graph: find the vertex of the parabola (or the corner of an absolute value), which gives the shifts; the stretch shows in one step to the right of the vertex; branches going down mean a minus in front.

The slope: how much the line rises per square to the right.

Where the line crosses the $y$-axis.

The stretch: how much the graph rises one step from the vertex. Negative means the branches go down.

The shift along $x$: the vertex's $x$.

The shift along $y$: the vertex's $y$.

Example: the vertex at $(1,\ 1)$, branches down, one step right goes down by $1$ — the formula is $y = -(x - 1)^2 + 1$.

Common mistakes

Mixing up the axes: $f(-1)$ is the height of the graph above $x = -1$, not the point where $y = -1$.
Dividing run by rise, or subtracting in a different order on top and at the bottom: both differences start from the same point.
The sign of a shift: a vertex at $x = 1$ means $(x - 1)^2$, not $(x + 1)^2$.
Missing a stretch: if one step right of the vertex the graph rises by $2$, the factor in front is $2$.