Identify and interpret key features of a graph, including x- and y-intercepts, intervals where the function is increasing or decreasing, relative maximums and minimums, and end behavior, in terms of a context (MA.912.F.1.3).

What this topic is asking

MA.912.F.1.3 asks you to read a graph's key features and say what they mean. The features are the intercepts, the intervals of increase and decrease, the relative maximum and minimum, and the end behavior. The B.E.S.T. Algebra 1 EOC almost always pairs the feature with a context ("what does the $y$-intercept represent?"), so you must both find it and interpret it.

Intercepts

• The $y$-intercept is the point where $x = 0$. In a context it is usually the starting value (initial height, initial cost, initial amount).
• The $x$-intercept (also called a zero or root) is where $y = 0$. In context it often marks when something runs out, lands, or breaks even.

A function can have one $y$-intercept (a function passes the vertical line test) but several $x$-intercepts.

Increasing, decreasing, maximum, minimum

Reading left to right, a graph is increasing where it goes up and decreasing where it goes down. A relative maximum is a high point (a peak) and a relative minimum is a low point (a valley). Crucially, the value of the max or min is the $y$-coordinate, while the $x$-coordinate tells you where it happens.

End behavior

End behavior describes the output as the input grows very large or very negative. A line rises or falls without bound. An upward parabola goes to $+\infty$ on both ends; a downward parabola goes to $-\infty$ on both ends. An exponential growth graph shoots up on the right and flattens toward a horizontal asymptote on the left. End behavior is how you tell function families apart at a glance.

How the B.E.S.T. EOC examines this topic

• Multiple choice. Interpret an intercept, a maximum, or a rate in a context.
• GRID and hot-spot. Click the maximum, an intercept, or select the increasing interval.
• Multiselect. Choose all true statements about a graph's features.

A clarifying idea: every key feature answers a specific question about the situation. The $y$-intercept answers "where did it start?", an $x$-intercept answers "when does it reach zero?", the maximum answers "what is the most?", and the increasing interval answers "while what is happening?". Translating the feature into the question keeps interpretations correct.

Why the output, not the input, is the maximum value

A frequent error is reporting the maximum as the $x$-coordinate of the peak. The maximum value of a function is the largest output, so it is the $y$-coordinate; the $x$-coordinate only tells you the input at which that largest output occurs. If $f(x)$ is height and the vertex is $(3, 16)$, the object is highest at time $x = 3$ seconds, and that greatest height is $16$ feet. The EOC builds a distractor out of the swapped value almost every time, so a useful habit is to phrase the answer as a sentence: "the maximum height of 16 feet occurs at 3 seconds," which forces you to attach the right number to the right quantity.

Connecting features to the equation

Many features can be found from the equation, which matters because the on-screen calculator is scientific, not graphing. The $y$-intercept is $f(0)$, found by substituting $x = 0$. The $x$-intercepts are the solutions of $f(x) = 0$, found by factoring or the quadratic formula. For a parabola in standard form, the vertex $x$-coordinate is $\frac{-b}{2a}$, and substituting gives the maximum or minimum value. Building these from the equation, rather than reading a screen, is exactly the skill the B.E.S.T. test design assumes.

Try this

Q1. A line has $y$-intercept $(0, -6)$ and $x$-intercept $(3, 0)$. If $y$ is a bank balance in dollars after $x$ weeks, what does the $x$-intercept mean? [1 point]

• Cue. The balance reaches $0 dollars after 3 weeks.

Q2. An upward parabola has vertex $(-2, -5)$. State the minimum value and where it occurs. [1 point]

• Cue. Minimum value $-5$, at $x = -2$.