10.3 Interpreting Key Features of Function Graphs: Intercepts, Increasing/Decreasing Intervals, Maximums/Minimums & End Behavior

Why Reading Function Graphs Matters on the GED

Assessment target A.5.e asks you to read a graph the way a scientist or business analyst would — not just plot points, but describe what is happening to the quantity the graph represents. These questions typically show a curve, often not a straight line, representing something like temperature over a day, a company's profit over the year, or a ball's height over time, and ask you to identify specific features: where it crosses zero, where it is rising or falling, where it peaks or bottoms out, and what happens to it far to the left or right.

Intercepts: Where a Graph Crosses the Axes

The y-intercept is the point where a graph crosses the y-axis — the value of the function when the input, x, equals 0. In a word problem it often represents a starting amount, such as the initial temperature at midnight or the starting balance of an account. The x-intercept(s), also called the zeros or roots of a function, are the point(s) where a graph crosses the x-axis, meaning the output is 0. In context, an x-intercept often represents when a quantity runs out, breaks even, or returns to a baseline.

A function can have at most one y-intercept, since one input (x = 0) can only produce one output, but it can have several x-intercepts — a parabola can cross the x-axis zero, one, or two times, and a wavier curve can cross it many times.

Increasing, Decreasing, Positive & Negative Intervals

Reading a graph from left to right (the direction x increases):

These four descriptions are independent of one another — a graph can be both decreasing and positive at the same time (falling, but still above the x-axis), so never assume "decreasing" automatically means "negative," or that "increasing" automatically means "positive."

Worked Scenario

A company's daily profit, in thousands of dollars, rises steadily from x = 0 to x = 6 hours, reaches its highest point at x = 6, then falls steadily from x = 6 to x = 10. The function is increasing on the interval 0 < x < 6 and decreasing on the interval 6 < x < 10.

Relative Maximums and Minimums

A relative maximum is a point that sits higher than every nearby point on the graph — the curve rises right up to that point, then falls immediately after. A relative minimum is the reverse: the curve falls right up to that point, then rises immediately after. On the GED, the most common source of these is a parabola's vertex: if the parabola opens upward, the vertex is a relative (and, for that curve, absolute) minimum; if it opens downward, the vertex is a relative maximum.

Worked Example

A quadratic function's graph has a vertex at (3, -4) and opens upward. Because the parabola opens upward, every other point on the graph sits higher than the vertex, so (3, -4) represents a relative minimum value of -4 at x = 3 — not a maximum, and not automatically the y-intercept (the y-intercept is wherever the graph crosses the y-axis, a different point entirely unless the vertex happens to sit at x = 0).

End Behavior

End behavior describes what happens to a function's output values as x moves toward positive infinity (far to the right of the graph) or negative infinity (far to the left). For the polynomial functions tested on the GED, end behavior is controlled by the leading term — the term with the highest exponent:

Worked Example

For f(x) = -2x² + 3x + 1, the leading term is -2x² — a negative coefficient on an even power. Both ends of the graph fall: as x becomes very large in either the positive or the negative direction, f(x) decreases without bound. This matches a downward-opening parabola, whose two arms both point down.

Symmetry

Some GED graphs are symmetric — a parabola, for example, is a mirror image of itself across a vertical line through its vertex called the axis of symmetry. Recognizing symmetry lets you find a matching point on one side of a graph once you already know a point on the other side, without recalculating from the equation.

GED Exam Scenario

A physics-flavored GED item graphs the height of a thrown ball over time as a downward-opening parabola. The ball starts at a height of 5 feet (the y-intercept), rises to a maximum height of 45 feet at 2 seconds (the vertex, a relative maximum), then falls back down, hitting the ground — an x-intercept — at 3 seconds. A follow-up question might ask for the interval during which the ball's height is increasing (0 < t < 2) or ask what the x-intercept at t = 3 represents in context (the moment the ball lands).

Common Traps

• Reading graph features right to left instead of left to right, which reverses "increasing" and "decreasing."
• Assuming "decreasing" means the same thing as "negative" (below the x-axis) — the two describe different features and can overlap or not.
• Mixing up the vertex of an upward-opening parabola (a minimum) with a downward-opening one (a maximum).
• Forgetting that end behavior describes only the far left and far right of a graph, not what happens in the middle, where a function can rise and fall several times before settling into its end behavior.