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Key Features of Graphs and Tables

Read and sketch intercepts, increasing and decreasing intervals, maximums and minimums, symmetry, end behavior and periods.

Practice this skill 22 min Key features of graphs and tables Leer en español
Key idea

What you'll learn

A graph tells a story. In this lesson you'll learn to read the key features of a graph or table — and to sketch a graph when you're told its features.

The key features are:

  • intercepts — where the graph crosses the axes;
  • increasing / decreasing intervals — where the graph goes up or down;
  • maximum / minimum points — the highest and lowest points (peaks and valleys);
  • symmetry — whether one side mirrors the other;
  • end behavior — what happens at the far left and far right;
  • periodicity — whether the graph repeats in cycles.

Why it matters on the GED. You'll see graphs of real situations — a ball's height, a day's temperatures, tides, profits — and must answer "What is the maximum?", "When is it decreasing?", "What does the intercept mean?", or click on a feature. These questions work for lines and for curves (nonlinear functions).

Real life. Finding the peak hour of traffic, the lowest price of a stock, when a medicine level is highest, or when a tank is empty are all key-feature questions.

Words to know

  • function — a rule that gives exactly one output y (or f(x)) for each input x.
  • x-intercept — a point where the graph crosses the x-axis; there y = 0 (e.g. (3, 0)). Also called a zero.
  • y-intercept — the point where the graph crosses the y-axis; there x = 0 (e.g. (0, -3)).
  • increasing — the graph goes up as you move left to right.
  • decreasing — the graph goes down as you move left to right.
  • interval — a stretch of x-values, such as 2 < x < 4.
  • maximum — the highest point (peak); the maximum value is its y-value.
  • minimum — the lowest point (valley); the minimum value is its y-value.
  • local (relative) max/min — highest/lowest point compared with nearby points only.
  • line of symmetry — a vertical line that the graph mirrors across (e.g. x = 1).
  • end behavior — what y does as x goes far right or far left (e.g. "rises without bound").
  • periodic — repeating the same pattern over and over; the period is the length of one cycle (e.g. 12 hours).

All Math words in the glossary

Intercepts, maximums and minimums

Graph of an upside-down U-shaped curve (parabola opening downward) on a coordinate plane from −5 to 5. Labeled key features: maximum point (−1, 4); x-intercepts (−3, 0) and (1, 0); y-intercept (0, 3); dashed line of symmetry x = −1. The graph increases for x < −1 and decreases for x > −1.−5−4−3−2−112345−5−4−3−2−1123450xymaximum (−1, 4)x-int (−3, 0)x-int (1, 0)y-int (0, 3)symmetry: x = −1

Intercepts.

  • y-intercept: where the graph meets the vertical axis. Set x = 0. A function has at most one.
  • x-intercepts: where the graph meets the horizontal axis. Set y = 0. There can be zero, one, or many.

In real life, intercepts are often "start" and "end" points. For a ball's height vs time, the t-intercepts are when the ball leaves the ground and when it lands.

Maximum and minimum.

  • A maximum is a peak: the graph rises to it, then falls.
  • A minimum is a valley: the graph falls to it, then rises.
  • Write the point as (x, y); the value is the y-coordinate. "The maximum height is 64 ft" (a y-value) is different from "the maximum occurs at 2 s" (an x-value).
  • A local max/min is a peak or valley compared with nearby points; the graph may go higher or lower elsewhere.

Example (graph below). The downward parabola has a maximum at (-1, 4), x-intercepts (-3, 0) and (1, 0), and y-intercept (0, 3).

Lines have no maximum or minimum (unless the domain is limited) — they rise or fall forever.

Increasing, decreasing and rate of change

Always read a graph from left to right, like a sentence.

  • Increasing on an interval: as x gets bigger, y gets bigger (the graph climbs).
  • Decreasing: as x gets bigger, y gets smaller (the graph falls).
  • Constant: a flat, horizontal piece.

Intervals are written with x-values. For the parabola above: increasing for x < -1, decreasing for x > -1. The switch happens at the maximum.

Graph of the downward parabola with maximum (−1, 4), x-intercepts (−3, 0) and (1, 0) and y-intercept (0, 3). The region x < −1 is shaded green and labelled increasing: the curve climbs from left to right. The region x > −1 is shaded red and labelled decreasing: the curve falls. The switch happens at the maximum, x = −1.−5−4−2234−5−4−3−2−11250xyincreasing: x < −1decreasing: x > −1max (−1, 4)(−3, 0)(1, 0)(0, 3)↗↘

In a table, look at the outputs as x increases:

x-3-2-10123
f(x)50-3-4-305

Outputs fall from 5 to -4 (decreasing for x < 0), then rise (increasing for x > 0). Minimum -4 at x = 0; x-intercepts at \pm 2; y-intercept -4.

The table x = −3 to 3 with f(x) = 5, 0, −3, −4, −3, 0, 5 is plotted as seven points on an upward parabola. Outputs fall until x = 0 (decreasing for x < 0) and rise after it (increasing for x > 0). The minimum is (0, −4), which is also the y-intercept, and the x-intercepts are (−2, 0) and (2, 0).x−3−2−10123f(x)50−3−4−305falls: 5 → −4then rises: −4 → 5−4−3−1134−5−3−2−1123460xymin (0, −4)(−2, 0)(2, 0)

How fast? On a line graph made of segments, the steepest rising segment is the fastest increase. Find each segment's change (or slope) and compare. A high point is not the same as a fast increase.

Linear vs nonlinear. A linear function increases (or decreases) at a constant rate: equal steps in x give equal changes in y. A nonlinear function's rate changes — the curve bends.

Symmetry, end behavior and periodicity

Symmetry. A graph is symmetric about a vertical line if folding along that line makes the two halves match. Parabolas are always symmetric about the vertical line through their max or min.

  • The line of symmetry is halfway between the x-intercepts: intercepts -3 and 1 → x = \frac{-3 + 1}{2} = -1.
  • Points the same distance from the line have the same y-value. You can use this to fill in missing table values.
  • Some graphs (like f(x) = x^3 - 3x) are symmetric about the origin: spin the graph half a turn and it looks the same.
The downward parabola with x-intercepts (−3, 0) and (1, 0). The line of symmetry x = −1 is halfway between them: (−3 + 1)/2 = −1, and each intercept is 2 units from it. Mirror points are the same distance from the line and have the same y-value: (−2, 3) and (0, 3).−5−4234−31250xyx = −122(−3, 0)(1, 0)(−2, 3)(0, 3)
Graph of f(x) = x³ − 3x on a plane from −3 to 3 by −4 to 4. It has a local maximum at (−1, 2) and a local minimum at (1, −2), and passes through the origin. It is symmetric about the origin: turning it half a turn about (0, 0) maps (−1, 2) onto (1, −2). Its left end goes down and its right end goes up.−3−223−4−3340xylocal max (−1, 2)local min (1, −2)originsymmetry

End behavior. Describe what y does at the far ends:

  • "As x increases without bound, y increases without bound" (the right end goes up).
  • "As x decreases without bound, y decreases without bound" (the left end goes down).
  • A downward parabola goes down on both ends; an upward parabola goes up on both ends; a line with positive slope goes down on the left and up on the right.
Three sketches of end behavior. Left: a downward parabola goes down on both ends. Middle: an upward parabola goes up on both ends. Right: a line with positive slope goes down on the left end and up on the right end.Downward parabolaleft end: downright end: downUpward parabolaleft end: upright end: upLine, positive slopeleft end: downright end: up

Periodicity. A periodic graph repeats the same shape: tides, seasons, a Ferris wheel's height. The period is the horizontal length of one cycle — measure peak to peak or valley to valley. Example: highs at 6 and 18 hours → period 18 - 6 = 12 hours.

Sketching from features. Plot the given intercepts and turning points first, then connect them, going up or down where the description says, and finish the ends according to the end behavior.

Formula

Formulas and rules

x\text{-intercept: set } y = 0 \qquad y\text{-intercept: set } x = 0 \text{line of symmetry (parabola)} = \frac{x_1 + x_2}{2} \quad \text{(average of the } x\text{-intercepts)} \text{turning point of } y = ax^2 + bx + c: \quad x = -\frac{b}{2a} \text{period} = \text{distance between two neighboring peaks}
  • x_1, x_2 = the x-intercepts; a, b, c = coefficients; a < 0 → maximum, a > 0 → minimum.

The GED formula sheet shows the standard form of a quadratic y = ax^2 + bx + c and the quadratic formula, but none of the rules above — memorize them. Most key-feature questions are answered by reading the graph or table, not by a formula.

Worked example

Example 1 — Read features from a graph

Problem. For the upward parabola below, give the x-intercepts, the minimum and the line of symmetry.

Graph of a U-shaped curve (parabola opening upward) on a coordinate plane from −5 to 5. It crosses the x-axis at (−1, 0) and (3, 0), crosses the y-axis at (0, −3), and its lowest point is (1, −4). It is symmetric about the vertical line x = 1.−5−4−3−2−112345−5−4−3−2−1123450xy
  1. The curve crosses the x-axis at x = -1 and x = 3 → (-1, 0) and (3, 0).
  2. The lowest point is (1, -4) → minimum value -4.
  3. Line of symmetry: x = \frac{-1 + 3}{2} = 1.
The upward parabola y = (x + 1)(x − 3) with key features labelled: x-intercepts (−1, 0) and (3, 0), y-intercept (0, −3), minimum (1, −4) on the dashed line of symmetry x = 1. Decreasing for x < 1 (red) and increasing for x > 1 (green).−3−2245−5−2−1123450xydecreasing: x < 1increasing: x > 1x = 1(−1, 0)(3, 0)(0, −3)min (1, −4)

Answer: intercepts (-1, 0), (3, 0); minimum (1, -4); symmetry x = 1.

Check: The minimum lies on the line of symmetry, x = 1 ✓

Worked example

Example 2 — Real world: ball height

Problem. A ball's height is graphed below. When does it land, what is its maximum height, and when is it rising?

Graph of a ball's height in feet (vertical axis, 0 to 80, labeled every 16) versus time in seconds (horizontal axis, 0 to 5). The curve starts at (0, 0), rises to its highest point (2, 64), and falls back to the ground at (4, 0). It is symmetric about t = 2.01234501632486480Time (seconds)Height (feet)
  1. It lands when height returns to 0: the second t-intercept, t = 4 s.
  2. Highest point: (2, 64) → maximum height 64 ft at 2 s.
  3. Rising (increasing) from t = 0 to t = 2.
Ball height h = −16t² + 64t in feet against time t in seconds. It leaves the ground at (0, 0), rises to the maximum (2, 64) and lands at (4, 0). The rising part 0 < t < 2 is shaded green; the falling part 2 < t < 4 is shaded red. The dashed line t = 2 is the line of symmetry.135163248800Time t (seconds)Height (ft)rising 0 < t < 2falling 2 < t < 4max (2, 64)lands (4, 0)6424

Answer: lands at 4 s; max 64 ft; rising for 0 < t < 2.

Check: The peak is halfway between 0 and 4 — symmetric ✓

Worked example

Example 3 — Real world: periodic tides

Problem. Use the tide graph. What are the period, the maximum and minimum depths, and when is the water rising in the morning?

Graph of water depth in feet (vertical axis, 0 to 10) at a harbor versus hours after midnight (horizontal axis, 0 to 24). The wave-shaped curve starts at its lowest depth 2 ft at t = 0, rises to its highest depth 8 ft at t = 6, falls to 2 ft at t = 12, rises to 8 ft at t = 18 and returns to 2 ft at t = 24. The pattern repeats every 12 hours.0246810121416182022240246810Hours after midnightWater depth (feet)
  1. Peaks at t = 6 and t = 18 → period = 12 hours.
  2. Maximum depth 8 ft; minimum depth 2 ft.
  3. From midnight (t = 0, low) to 6 a.m. (t = 6, high) the depth increases.
Tide depth in feet against hours after midnight, from 0 to 24 hours. The wave has lows of 2 ft at t = 0, 12 and 24 and highs of 8 ft at t = 6 and 18. An arrow from peak to peak shows the period, 18 − 6 = 12 hours. The rising stretch 0 < t < 6 is shaded green. Dashed lines mark the maximum 8 ft and minimum 2 ft.612182446100Hours after midnight (t)Depth (ft)rising0 < t < 682period = 18 − 6 = 12 hmax 8 ftmin 2 ft

Answer: period 12 h; max 8 ft, min 2 ft; rising 0 < t < 6.

Check: Valleys at t = 0 and t = 12 also give 12 - 0 = 12 ✓

Worked example

Example 4 — Work backwards with symmetry

Problem. A parabola has x-intercepts (-5, 0) and (1, 0), and passes through (0, -5). Find its line of symmetry, and another point with y = -5.

  1. Line of symmetry: x = \frac{-5 + 1}{2} = -2.
  2. (0, -5) is 2 units right of x = -2.
  3. Its mirror point is 2 units left: x = -2 - 2 = -4 → (-4, -5).
The parabola y = (x + 5)(x − 1) with x-intercepts (−5, 0) and (1, 0). Its line of symmetry is x = −2, halfway between. The point (0, −5) is 2 units right of the line; its mirror point (−4, −5) is 2 units left, at the same height. The minimum is (−2, −9).−7−6−3−123−10−8−4−3−2−11230xyx = −222(−5, 0)(1, 0)(0, −5)(−4, −5)min (−2, −9)

Answer: symmetry line x = -2; the point (-4,\ -5) is also on the graph.

Check: The function is y = (x + 5)(x - 1): at x = -4, (1)(-5) = -5 ✓

Worked example

Example 5 — GED-hard: max from an equation

Problem. A company's daily profit is P(x) = -2x^2 + 40x - 150 dollars, where x is the number of items made. How many items give the maximum profit, and what is it?

  1. a = -2 < 0, so the graph opens down and has a maximum.
  2. x = -\frac{b}{2a} = -\frac{40}{2(-2)} = 10 items.
  3. P(10) = -2(100) + 40(10) - 150 = -200 + 400 - 150 = 50.
Graph of daily profit P(x) = −2x² + 40x − 150 dollars against items made x, from 0 to 20. The parabola opens down. It crosses zero profit at x = 5 and x = 15, and its maximum is (10, 50), on the line x = −b/(2a) = 10. Nearby points (9, 48) and (11, 48) are equal and lower on both sides.51520−50−40−30−20−10102030405060Items made (x)Profit P ($)x = −40/(2(−2)) = 10max (10, 50)(9, 48)(11, 48)(5, 0)(15, 0)

Answer: 10 items give the maximum profit of $50.

Check: P(9) = -162 + 360 - 150 = 48 and P(11) = -242 + 440 - 150 = 48 — equal and lower on both sides ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

  • Most key-feature questions are read from the graph — no calculator needed.
  • To evaluate a function at a point, type it with parentheses and the x² key: -16(2.5)^2 + 80(2.5) is (−) 16 × 2.5 x² + 80 × 2.5 enter → 100.
  • Use the (−) key for the leading negative, not the subtraction key.
  • To find a turning point, compute -\frac{b}{2a} with the n/d key, or evaluate the function at several nearby x-values and look for where outputs stop rising and start falling.
  • When NOT to use it: reading intercepts, intervals, period or symmetry from a picture.
Common mistake

Common mistakes

  • Mixing up x and y answers. "Maximum height" is a y-value (64 ft), not the time it happens (2 s). ✗ Re-read whether the question asks "what" (output) or "when/where" (input). ✓
  • Confusing the intercepts. Giving f(x) = 0 as the y-intercept. ✗ f(x) = 0 gives x-intercepts; the y-intercept is f(0). ✓
  • Describing intervals with y-values. "Decreasing from 64 to 0." ✗ Use x-values: decreasing for 2 < t < 4. ✓
  • Stretching intervals past the turning points. Saying f decreases for -2 < x < 2 when it only falls between -1 and 1. ✗ Intervals change exactly at a max or min. ✓
  • Highest ≠ fastest. The peak (70°F) is not where temperature rose fastest. ✗ Fastest increase = steepest rising segment. ✓
  • Measuring the period vertically. The peak height (8 ft) is not the period. ✗ Period is a horizontal distance: peak to peak (12 h). ✓
Test strategy

GED test strategy

  • Trace with your finger (or the mouse pointer) from left to right, saying "up… peak… down…" out loud in your head. Mark turning points on your erasable note board.
  • Check the axes: labels, units and scale. Many traps come from reading the wrong axis.
  • Eliminate by direction. For increasing/decreasing questions, test one x-value inside each answer's interval — if the graph goes the wrong way there, cross it out.
  • Use symmetry to check answers: max/min should be halfway between the x-intercepts of a parabola.
  • For select-all questions, judge each statement separately as true/false; don't assume a fixed number of correct answers.
  • When an equation is given, a quick table of values (3–5 points) often answers the question faster than algebra.

Check yourself

Key features of graphs and tablesNo calculator

What is the y-intercept of the graph shown?

Graph of a U-shaped curve (parabola opening upward) on a coordinate plane from −5 to 5. It crosses the x-axis at (−1, 0) and (3, 0), crosses the y-axis at (0, −3), and its lowest point is (1, −4). It is symmetric about the vertical line x = 1.−5−4−3−2−112345−5−4−3−2−1123450xy
Answer choices
Hint

The y-intercept is where the graph crosses the vertical axis.

Check yourself

Key features of graphs and tablesNo calculatorHot spot

Click the minimum point of the graph.

Click on the figure to mark your answer (or focus it and use the arrow keys).

Graph of a U-shaped curve (parabola opening upward) on a coordinate plane from −5 to 5. It crosses the x-axis at (−1, 0) and (3, 0), crosses the y-axis at (0, −3), and its lowest point is (1, −4). It is symmetric about the vertical line x = 1.−5−4−3−2−112345−5−4−3−2−1123450xy
Hint

The minimum is the lowest point on the whole graph — the bottom of the U.

Summary

  • y-intercept: set x = 0 (only one). x-intercepts: set y = 0 (can be several).
  • Increasing = graph rises left to right; decreasing = falls. Give intervals in x-values; they switch at a max or min.
  • Maximum/minimum = peak/valley; the value is the y-coordinate. Parabolas: a < 0 → max, a > 0 → min, at x = -\frac{b}{2a}.
  • Symmetry: a parabola's line of symmetry is halfway between its x-intercepts; mirror points share a y-value.
  • End behavior describes the far left and far right; periodic graphs repeat, with period = peak-to-peak distance.
  • Read the question twice: "what" asks for an output, "when/where" asks for an input.
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