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Comparing Functions in Different Forms

Find the rate of change, intercepts, and maximum or minimum of linear and quadratic functions from equations, tables, and graphs, and compare functions.

Practice this skill 30 min Compare functions in different forms Leer en español
Key idea

What you'll learn

Two functions can be shown in completely different ways — one as an equation, the other as a graph, a table, or a sentence. In this lesson you will learn to pull the same key properties out of each form so you can compare them fairly:

  • Rate of change (slope) of a linear function
  • y-intercept (the starting value) and x-intercepts (the zeros)
  • Maximum or minimum (the vertex) of a quadratic function
  • Whether a function is increasing or decreasing, and where

Why it matters on the GED. Typical questions: "Function f is given by an equation and function g by a graph — which has the greater rate of change?", "Which function has the greater maximum value?", "Taxi A's fare is an equation, Taxi B's fares are in a table — which statement is true?". These questions test whether you understand what the numbers mean, not just how to compute them.

Why it matters in real life. Comparing phone plans, gym memberships, savings accounts, or the flight of two rockets all come down to the same few properties: where do you start, how fast do you change, and what is the highest (or lowest) value?

Words to know

  • Rate of change — how much the output changes for each 1-unit increase in the input; for a line it is the slope m. (f(x) = 3x + 2 has rate 3.)
  • y-intercept — where the graph crosses the y-axis; the output when x = 0, i.e. f(0). (f(x) = 3x + 2 → (0, 2).)
  • x-intercept (zero, root) — where the graph crosses the x-axis; an input with f(x) = 0. (x^2 - 4 has zeros 2 and -2.)
  • Increasing / decreasing — the graph goes up / down as you move left to right.
  • Parabola — the U-shaped graph of a quadratic f(x) = ax^2 + bx + c.
  • Vertex — the turning point of a parabola; it is the maximum if the parabola opens down (a < 0) or the minimum if it opens up (a > 0).
  • Maximum / minimum value — the y-coordinate of the vertex (the output), not the x-coordinate.
  • Axis of symmetry — the vertical line x = h through the vertex; the parabola is a mirror image on either side.
  • Starting value — in a real-world function, the output at time 0 (a fee, a deposit, a launch height).

All Math words in the glossary

1. Properties of linear functions in every form

A linear function f(x) = mx + b has rate of change m and y-intercept b. Here is how to find them in each form:

FormRate of change my-intercept b
Equation y = mx + bthe number multiplying xthe constant term
Table\frac{\text{change in } y}{\text{change in } x} between two rowsthe y-value where x = 0 (or work back to x = 0 using the rate)
Graph\frac{\text{rise}}{\text{run}} between two clear pointswhere the line crosses the y-axis
Descriptionthe "per" amount ($2 per mile)the one-time amount (a $4 fee)

Example — table. x: 2, 4, 6 and y: 8, 12, 16. Rate: \frac{12 - 8}{4 - 2} = \frac{4}{2} = 2. Work back to x = 0: going from x = 2 to x = 0 is 2 steps left, so subtract 2 \times 2 = 4: 8 - 4 = 4. So y = 2x + 4.

Table with x = 2, 4, 6 and y = 8, 12, 16, plus a green row x = 0, y = 4 found by working back. Beside it, the graph of y = 2x + 4 through (2, 8), (4, 12) and (6, 16). A red step from (2, 8) runs 2 and rises 4 to (4, 12), so the rate is 4 ÷ 2 = 2. The line crosses the y-axis at (0, 4), the y-intercept, so it does not pass through the origin.xy0428412616y = 2x + 4green row: workedback to x = 001234567048121620xyrun 2rise 4(2, 8)(4, 12)(6, 16)(0, 4): y-interceptrate = 4 ÷ 2 = 2

Watch out: \frac{8}{2} = 4 is not the rate here, because the line does not pass through the origin. Use changes in y over changes in x.

Example — two points (no table). A line through (-2, 9) and (4, -3): m = \frac{-3 - 9}{4 - (-2)} = \frac{-12}{6} = -2. Then 9 = -2(-2) + b \Rightarrow b = 5.

Graph of the line through (−2, 9) and (4, −3). A red step runs 6 to the right and drops 12, so the slope is m = −12 ÷ 6 = −2. The line crosses the y-axis at (0, 5), so b = 5 and the equation is y = −2x + 5.−4−2246−6−4−2246810120xyrun = 6rise = −12(−2, 9)(4, −3)(0, 5): b = 5m = −12 ÷ 6 = −2y = −2x + 5

Comparing rates. A larger positive slope means the function grows faster. For negative slopes, compare sizes: a slope of -3 decreases faster than a slope of -2, because it drops 3 units per step instead of 2.

x-intercept of a line: set f(x) = 0. For f(x) = 2x - 6: 2x - 6 = 0 \Rightarrow x = 3.

2. Properties of quadratic functions in every form

A quadratic f(x) = ax^2 + bx + c has a U-shaped graph.

Graph of y = x squared + 2x - 3, an upward-opening parabola. Labeled features: the vertex (-1, -4) is the minimum point; the x-intercepts are (-3, 0) and (1, 0); the y-intercept is (0, -3).−7−6−5−4−3−2−11234−5−4−3−2−112345xyvertex (−1, −4) = minimumx-intercept (−3, 0)x-intercept (1, 0)y-intercept (0, −3)

For f(x) = x^2 + 2x - 3 (shown): it opens up (a = 1 > 0), so its vertex (-1, -4) is a minimum; the minimum value is -4. The y-intercept is f(0) = -3; the x-intercepts are -3 and 1.

From an equation f(x) = ax^2 + bx + c:

  1. Direction: a > 0 opens up (minimum); a < 0 opens down (maximum).
  2. Vertex x-coordinate: x = -\dfrac{b}{2a}.
  3. Max/min value: plug that x back in: f\left(-\dfrac{b}{2a}\right).
  4. y-intercept: c.
  5. x-intercepts: solve f(x) = 0 by factoring or the quadratic formula.

Example: f(x) = -x^2 + 6x - 4: a = -1 (opens down → maximum). x = -\frac{6}{2(-1)} = 3. Max value f(3) = -9 + 18 - 4 = 5.

From a graph: read the highest or lowest point directly; read where the curve crosses each axis.

From a table: use symmetry. Equal outputs sit at equal distances from the vertex. In the table below, f(1) = f(3) = -1, so the vertex is halfway between, at x = 2, and the minimum value is f(2) = -2.

x-10123
f(x)72-1-2-1

From a description: words like "reaches its highest point of 144 feet after 3 seconds" give the vertex (3, 144) directly; "starts at 24 feet" gives the y-intercept.

Increasing and decreasing. A parabola that opens up decreases to the left of its vertex and increases to the right of it; a parabola that opens down does the opposite. For f(x) = x^2 + 2x - 3 (vertex at x = -1), f decreases for x < -1 and increases for x > -1. A line with a positive slope always increases; a line with a negative slope always decreases.

3. Comparing two functions step by step

  1. Identify the property the question asks about (rate? y-intercept? max value? where the max happens?).
  2. Find that property for function 1, in its own form.
  3. Find the same property for function 2, in its own form.
  4. Compare — and answer in the words of the question (which one, by how much, or when).

Do not mix up "where" and "how high." The vertex (3, 5) means the maximum value is 5, and it occurs at x = 3. "Which function has the greater maximum?" compares 5-type numbers, not 3-type numbers.

Real-world meanings.

PropertyIn a word problem it often means…
rate of changeprice per mile, dollars saved per month, speed
y-interceptstarting fee, opening balance, launch height
x-interceptbreak-even point, when a ball lands
maximumgreatest height, greatest profit
minimumlowest cost, lowest temperature

When do two linear functions become equal? Set them equal and solve. Account A: 200 + 50m; Account B: 500 + 30m. 200 + 50m = 500 + 30m \Rightarrow 20m = 300 \Rightarrow m = 15. After 15 months A catches up, because its rate is larger even though its start is smaller.

Graph of two savings accounts, balance in dollars against months. Account A, 200 + 50m, starts at 200 dollars and rises faster (solid blue). Account B, 500 + 30m, starts at 500 dollars and rises more slowly (dashed red). The lines cross at (15, 950): after 15 months A catches up, and after that A is higher.0510152025030060090012001500Months (m)Balance in dollars200500(15, 950)A: 200 + 50m (smaller start, faster rate)B: 500 + 30m (bigger start, slower rate)
Formula

Formulas

m = \frac{y_2 - y_1}{x_2 - x_1} \qquad y = mx + b \qquad x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \qquad x_{\text{vertex}} = -\frac{b}{2a}
  • m — slope (rate of change) from two points (x_1, y_1) and (x_2, y_2)
  • b in y = mx + b — the y-intercept
  • a, b, c — coefficients of f(x) = ax^2 + bx + c; the quadratic formula gives the x-intercepts
  • x_{\text{vertex}} — the x-coordinate of the vertex; the max/min value is f(x_{\text{vertex}})

Formula sheet: the slope formula, slope-intercept form y = mx + b, the standard form y = ax^2 + bx + c, and the quadratic formula are on the GED formula sheet. The vertex formula x = -\frac{b}{2a} is not on the sheet — memorize it (notice it is the "\frac{-b}{2a}" part of the quadratic formula).

Worked example

Example 1 (easy): equation vs. graph (linear)

f(x) = 3x + 2. The function g is graphed below. Compare their rates of change and y-intercepts.

Graph of the line y = g(x) on a coordinate plane from -5 to 5. The line rises from left to right and passes through the marked points (0, -2) and (3, 4).−5−4−3−2−112345−5−4−3−2−112345xy(0, −2)(3, 4)y = g(x)
  1. f: rate = 3, y-intercept = 2.
  2. g: the line passes through (0, -2) and (3, 4). Rate = \frac{4 - (-2)}{3 - 0} = \frac{6}{3} = 2. y-intercept = -2.
  3. Compare: 3 > 2 and 2 > -2.

Answer: f has both the greater rate of change (3 vs. 2) and the greater y-intercept (2 vs. -2).

Check: g(x) = 2x - 2: g(3) = 6 - 2 = 4 ✓ matches the graph.

Worked example

Example 2: table vs. equation (gym memberships)

Gym A charges C(m) = 25 + 30m dollars for m months. Gym B's total costs are in the table. Compare the sign-up fees and monthly rates. When do the two gyms cost the same?

Months135
Total cost ($)70110150
  1. Gym B's rate: \frac{110 - 70}{3 - 1} = \frac{40}{2} = \text{\textdollar}20 per month.
  2. Gym B's fee: work back from 1 month to 0 months: 70 - 20 = \text{\textdollar}50. So B(m) = 50 + 20m.
  3. Gym A: fee $25, rate $30 per month. A has the lower fee; B has the lower monthly rate.
  4. Equal cost: 25 + 30m = 50 + 20m \Rightarrow 10m = 25 \Rightarrow m = 2.5.
Gym B's table (1 month 70 dollars, 3 months 110 dollars, 5 months 150 dollars) beside a graph of total cost against months. Gym B, B(m) = 50 + 20m, is a solid blue line starting at 50 dollars through the table points. Gym A, C(m) = 25 + 30m, is a dashed red line starting at 25 dollars that rises faster. The lines cross at (2.5, 100): before 2.5 months Gym A is cheaper, after it Gym B is cheaper.MonthsGym B ($)17031105150Gym B's tableB(m) = 50 + 20mGym A:C(m) = 25 + 30m0123456050100150200Months (m)Total cost in dollars(1, 70)(3, 110)(5, 150)25(2.5, 100)Left of 2.5: A cheaper. Right of 2.5: B cheaper.

Answer: Gym A is cheaper for 1 or 2 months; from 3 months on, Gym B is cheaper. They would be equal at 2.5 months.

Check: 3 months: A = 25 + 90 = \text{\textdollar}115, B = \text{\textdollar}110 ✓ B cheaper.

Worked example

Example 3: comparing maximums (equation vs. graph)

f(x) = -2x^2 + 8x + 1. The graph of g is shown. Which function has the greater maximum value?

Graph of y = g(x), a downward-opening parabola on a coordinate plane from -2 to 6 horizontally and -3 to 7 vertically. Its highest point (vertex) is (2, 6). Other marked points: (0, 2), (1, 5), (3, 5), and (4, 2).−2−1123456−3−2−11234567xy(2, 6)y = g(x)
  1. f: a = -2 < 0, so it has a maximum. Vertex x = -\frac{8}{2(-2)} = -\frac{8}{-4} = 2.
  2. f(2) = -2(2)^2 + 8(2) + 1 = -8 + 16 + 1 = 9. Max value of f: 9.
  3. g: the highest point on the graph is (2, 6). Max value of g: 6.
  4. Compare: 9 > 6.

Both functions on one set of axes:

Both functions on one coordinate plane. f(x) = −2x² + 8x + 1 (solid blue) opens down with highest point (2, 9) and passes through (1, 7) and (3, 7). g (dashed red, from the given graph) opens down with highest point (2, 6). A dotted vertical line x = 2 passes through both vertices; a green double arrow between them shows f’s maximum is 3 greater.−112345−4−22468100xy(2, 9)(2, 6)3(1, 7)(3, 7)f(x) = −2x² + 8x + 1g (from the graph)max of f = 9max of g = 6difference: 9 − 6 = 3

Answer: f has the greater maximum value, by 9 - 6 = 3. (Both maximums occur at x = 2.)

Check: f(1) = -2 + 8 + 1 = 7 and f(3) = -18 + 24 + 1 = 7 — equal, so x = 2 is indeed the axis of symmetry ✓

Worked example

Example 4: comparing minimums (table vs. equation)

Function f is given by the table. g(x) = x^2 + 4x + 1. Which has the smaller minimum value, and where does each minimum occur?

x-10123
f(x)72-1-2-1
  1. f: f(1) = f(3) = -1, so by symmetry the vertex is at x = 2; minimum value f(2) = -2.
  2. g: a = 1 > 0 (minimum). x = -\frac{4}{2(1)} = -2. g(-2) = 4 - 8 + 1 = -3.
  3. Compare values: -3 < -2.
Table of f: x = −1, 0, 1, 2, 3 gives f(x) = 7, 2, −1, −2, −1; since f(1) = f(3) = −1, the vertex is at x = 2. Beside it, both parabolas on one plane. f (solid blue, through the table points) has its lowest point at (2, −2). g(x) = x² + 4x + 1 (dashed red) has its lowest point at (−2, −3) and passes through (−3, −2) and (−1, −2). The minimum of g, −3, is smaller than the minimum of f, −2.xf(x)−17021−12−23−1f (table)f(1) = f(3) → vertex x = 2−5−4−3−2−112345−4−224680xy(2, −2)(−2, −3)(−3, −2)f: min −2 at x = 2g: min −3 at x = −2

Answer: g has the smaller minimum value, -3 (at x = -2); f's minimum is -2 (at x = 2).

Check: g(-1) = 1 - 4 + 1 = -2 and g(-3) = 9 - 12 + 1 = -2 — equal on both sides of x = -2 ✓

Worked example

Example 5 (GED-hard): two rockets

Rocket A's height is graphed. Rocket B's height, in feet, after t seconds is h(t) = -16t^2 + 80t + 24. Which rocket goes higher, by how much, and which reaches its peak first?

Graph of the height of Rocket A. Horizontal axis: Time in seconds, 0 to 6. Vertical axis: Height in feet, 0 to 160 in steps of 20. The curve is a downward-opening parabola that starts at (0, 0), passes through (1, 80), (2, 128), reaches its highest point at (3, 144), then passes through (4, 128), (5, 80), and lands at (6, 0).012345620406080100120140160Time (seconds)Height (feet)(1, 80)(3, 144)(5, 80)
  1. Rocket A: highest point on the graph (3, 144) → max 144 ft at 3 s.
  2. Rocket B: vertex t = -\frac{80}{2(-16)} = \frac{80}{32} = 2.5 s.
  3. h(2.5) = -16(6.25) + 80(2.5) + 24 = -100 + 200 + 24 = 124 ft.
  4. Compare: 144 - 124 = 20 ft; 2.5 < 3.

Both rockets on one set of axes:

Both rockets on one graph of height in feet against time in seconds. Rocket A (solid blue) starts at the ground (0, 0), peaks at (3, 144) and lands at 6 seconds. Rocket B, h(t) = −16t² + 80t + 24 (dashed red), starts at (0, 24), peaks at (2.5, 124) and lands a little after 5 seconds. A green double arrow shows that A’s peak is 20 feet higher, while dotted lines show B peaks earlier, at 2.5 s instead of 3 s.0123456020406080100120140160Time in seconds (t)Height in feet(3, 144)(2.5, 124)(0, 24)20 ftA (graph)B: h(t)

Answer: Rocket A goes 20 feet higher, but Rocket B reaches its peak first (2.5 s vs. 3 s). Note also B starts at 24 ft (h(0) = 24) while A starts at ground level.

Check: h(2) = -64 + 160 + 24 = 120 and h(3) = -144 + 240 + 24 = 120 — symmetric about 2.5 ✓

Calculator tip

TI-30XS MultiView tips

  • Slope from two points: type the whole fraction with the n/d key: numerator (−) 3 − 9, then denominator 4 − (−) 2, enter → −2.
  • Vertex x: compute -\frac{b}{2a} with parentheses around the denominator: (−) 80 ÷ ( 2 × (−) 16 ) enter → 2.5.
  • Max/min value: substitute the vertex x into the function, using x² for the square and parentheses for negatives.
  • Decimals vs. fractions: press toggle to switch between \tfrac{5}{2} and 2.5.
  • When NOT to use it: reading slopes and vertices from a clear graph — the answer is already in front of you. Do the reasoning ("which property?") on paper first.
Common mistake

Common mistakes

  • Using a ratio instead of a rate of change. ✗ Table (2, 8): "rate = 8 \div 2 = 4." ✓ Use changes: \frac{12 - 8}{4 - 2} = 2. \frac{y}{x} works only when the line goes through the origin.
  • Reporting the vertex's x as the max/min. ✗ "f's maximum is at x = 3, so its max is 3." ✓ The max value is f(3), the y-coordinate.
  • Comparing y-intercepts instead of maximum or minimum values. ✗ "g starts higher (18 vs. 10), so it has the higher minimum." ✓ The y-intercept is only f(0); find the vertex.
  • Sign slip in -\frac{b}{2a}. ✗ For x^2 - 10x + 29, x = -5. ✓ x = -\frac{-10}{2} = 5.
  • Reading a non-zero row as the y-intercept. ✗ "The first row is (-1, 1), so the y-intercept is 1." ✓ The y-intercept is the output at x = 0.
  • Comparing negative slopes by sign only. A slope of -3 is "less" than -2, but the function with slope -3 decreases faster.
Test strategy

GED test strategy

  • Underline the property the question names (rate, intercept, maximum value, where the maximum occurs). Many wrong choices compare the wrong property correctly.
  • Make a mini-table on your erasable note board: two columns (function 1, function 2) and rows for slope, y-int, vertex. Fill in only what you need.
  • Use the formula sheet for the slope formula and the quadratic formula; memorize x = -\frac{b}{2a}.
  • Check one more point: after you find a slope or vertex, verify it against another table row or graph point.
  • Eliminate statements with a correct conclusion but a wrong reason — on the GED the reasoning in the answer choice must also be true.
  • Inequality word problems ("first month when…"): test the whole numbers on both sides of your answer.

Check yourself

Compare functions in different formsNo calculator

The function f is defined by f(x) = 3x + 2. The function g is graphed below.

Graph of the line y = g(x) on a coordinate plane from -5 to 5. The line rises from left to right and passes through the marked points (0, -2) and (3, 4).−5−4−3−2−112345−5−4−3−2−112345xy(0, −2)(3, 4)y = g(x)

Which function has the greater rate of change, and what is that rate?

Answer choices
Hint

For g, use two marked points and compute rise ÷ run. For f, which number multiplies x?

Check yourself

Compare functions in different formsNo calculatorFill in

The linear function f is shown in the table. The function g is defined by g(x) = 2x + 9.

x0246
f(x)5111723

How much greater is the rate of change of f than the rate of change of g?

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Rate of change in a table: change in f(x) divided by change in x. Don't confuse the rate of g with its y-intercept.

Summary

  • Linear: rate of change = slope m (from \frac{\Delta y}{\Delta x}, rise/run, or the "per" amount); y-intercept = output at x = 0 (the starting value or fee).
  • Quadratic: a > 0 → minimum, a < 0 → maximum; vertex at x = -\frac{b}{2a}; the max/min value is the y-coordinate.
  • In a table, use symmetry (equal outputs) to locate the vertex; in a graph, read it directly.
  • To compare, find the same property for both functions, then compare the numbers — and answer in the question's words.
  • Two linear functions are equal where m_1x + b_1 = m_2x + b_2; the larger rate eventually wins.
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