A graph tells a story at a glance, if you know where to look. You know the domain and range; now name the other features people ask about.
- $x$-intercepts (also called zeros): where the graph crosses the $x$-axis, so $y = 0$.
- $y$-intercept: where it crosses the $y$-axis, so $x = 0$. Its height is $f(0)$.
- Increasing on an interval: as you move right, the graph goes up. Decreasing: as you move right, it goes down.
- Maximum and minimum: the highest and lowest outputs, at a peak or a valley.
Why it works
Graphs are always read from left to right, the direction in which $x$ grows. “Increasing” means that when the input gets bigger, the output gets bigger too. That is a statement about a stretch of inputs, so the answer is an interval of $x$-values, such as $(1, \infty)$, not a single point. A maximum value is an output, a $y$-coordinate; the point where it happens has both coordinates.
Key idea
Intercepts: set the other variable to $0$. Increasing or decreasing: read left to right and answer with an interval of $x$-values. Maximum or minimum value: the $y$-coordinate of the highest or lowest point.
Worked examples
Worked example: Reading the features
For $f(x) = -(x - 1)^2 + 4$ (graphed above), name its zeros, $y$-intercept, maximum and where it increases.
- Zeros: the graph crosses the $x$-axis at $x = -1$ and $x = 3$. Check: $f(3) = -(2)^2 + 4 = 0$.
- $y$-intercept: $f(0) = -(-1)^2 + 4 = 3$, the point $(0, 3)$.
- Maximum: the peak is $(1, 4)$, so the maximum value is $4$.
- Reading left to right, the graph rises until $x = 1$: increasing on $(-\infty, 1)$, decreasing on $(1, \infty)$.
Answer: Zeros $-1$ and $3$; $y$-intercept $3$; maximum $4$; increasing on $(-\infty, 1)$.
Worked example: Intercepts from a rule
Find the intercepts of $g(x) = 2x - 6$.
- $y$-intercept: $g(0) = -6$, the point $(0, -6)$.
- $x$-intercept: solve $2x - 6 = 0$, so $x = 3$, the point $(3, 0)$.
Answer: $(0, -6)$ and $(3, 0)$.
Common mistake
Describing “increasing” by reading right to left, or giving a point when an interval is asked for. Always move left to right, and answer with the $x$-values where the rise happens. Check by picking two inputs in your interval: the larger input should give the larger output.