Interpret and Evaluate Linear and Quadratic Functions - GED Math

SECTION 1

Why Functions Matter — A Brief History

Long before algebra had a name, people needed ways to describe how one quantity depends on another. A farmer notices that doubling her land doubles her harvest — that is a linear relationship. A stone thrown upward traces a curved arc through the air — that path follows a quadratic relationship. The idea of a function, a rule that assigns exactly one output to each input, took centuries to develop into the clear notation we use today.

~300 BCE • Euclid's Proportions

Greek mathematicians studied proportional relationships — the idea that doubling one quantity doubles another — laying the groundwork for linear thinking.

~825 CE • Al-Khwarizmi's Algebra

The Persian mathematician al-Khwarizmi published methods for solving equations involving squares (x²), giving rise to the word 'algebra' and to quadratic techniques still used today.

1637 • Descartes' Coordinate System

René Descartes introduced the x-y coordinate plane, making it possible to draw equations as graphs — lines, parabolas, and other curves became visible for the first time.

1748 • Euler Formalizes Functions

Leonhard Euler popularized f(x) notation, giving us the precise language we use on the GED today: plug in a value for x and get a single output.

On the GED, roughly 55 percent of the math test focuses on algebraic problem solving, and functions are at the heart of it. You will be asked to evaluate a function at a given input, interpret what a graph or table tells you, and compare linear growth with quadratic growth. This lesson builds those skills from the ground up.

SECTION 2

Core Principles — What Is a Function?

A function is a rule that takes an input (often called x) and produces exactly one output (often called f(x) or y). Think of it as a machine: you feed in a number, the machine does its work, and one number comes out. On the GED, you will encounter two main types of functions.

Linear Function

Has the form f(x) = mx + b. The graph is a straight line. The rate of change (slope) is constant — every equal step in x produces the same change in y.

Quadratic Function

Has the form f(x) = ax² + bx + c. The graph is a U-shaped curve called a parabola. The rate of change increases or decreases — the curve gets steeper as you move away from the vertex.

Evaluating a Function

To evaluate f(3), replace every x in the formula with 3, then simplify. The result is the y-value at that point. This is the single most common function question on the GED.

Interpreting a Function

Interpreting means explaining what a function's parts mean in a real-world context. For example, in C(x) = 15x + 200, the 15 is the cost per item and 200 is the fixed setup cost.

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KEY TAKEAWAY

Think of a function like a vending machine. You press a button (the input x), and exactly one item comes out (the output y). A linear function is like a machine that adds the same amount each time — predictable and steady. A quadratic function is like a machine that speeds up or slows down — the changes themselves change.

SECTION 3

Seeing the Difference — Linear vs. Quadratic Graphs

The cyan straight line represents f(x) = x + 1, which rises at a constant rate. The pink curve represents g(x) = x², which starts slowly and then rises faster and faster — the hallmark of a quadratic function.

Notice the key visual difference in the diagram above. The linear function climbs at the same rate from left to right — each step of 1 in x adds the same amount to y. The quadratic function starts nearly flat but then curves sharply upward. This accelerating pattern is why a ball thrown in the air doesn't just float — gravity pulls it back along a parabolic path. On the GED, you can identify a function as linear if its graph is a straight line and quadratic if its graph is a U-shape (or upside-down U).

SECTION 4

The Mathematical Framework

Linear Functions

SLOPE-INTERCEPT FORM

f(x) = mx + b

m = slope (rate of change — how much y changes for each 1-unit increase in x). b = y-intercept (the value of y when x = 0). This form is provided on the GED formula sheet.

SLOPE FORMULA

m = (y₂ − y₁) / (x₂ − x₁)

Given two points (x₁, y₁) and (x₂, y₂), the slope is the change in y divided by the change in x. This formula is also on the GED formula sheet.

To evaluate a linear function, substitute the given x-value into the formula and simplify. For example, if f(x) = 3x + 5, then f(4) = 3(4) + 5 = 12 + 5 = 17. To interpret a linear function in context, the slope m tells you the rate of change (like dollars per hour, miles per gallon), and the y-intercept b tells you the starting value.

Quadratic Functions

STANDARD FORM

f(x) = ax² + bx + c

a determines direction (positive a = opens upward, negative a = opens downward) and width of the parabola. c = y-intercept. The vertex (highest or lowest point) occurs at x = −b / (2a).

QUADRATIC FORMULA

x = (−b ± √(b² − 4ac)) / (2a)

Finds the x-values where f(x) = 0 (the x-intercepts or zeros). This formula is on the GED formula sheet. The expression under the square root, b² − 4ac, is called the discriminant and tells you how many solutions exist.

To evaluate a quadratic, follow the same substitution process but remember order of operations: square before you multiply, multiply before you add. For f(x) = 2x² − 3x + 1, evaluating f(−2) means: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15.

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GED TIP — FORMULA SHEET

You do not need to memorize the slope formula, the quadratic formula, or the standard forms — they are provided on the on-screen formula sheet. Focus your energy on knowing when and how to use each formula, not on memorizing them.

SECTION 5

Key Features — Slope, Vertex, and Intercepts

The GED frequently asks you to identify and interpret key features of function graphs and equations. The table below organizes these features side by side so you can see how the same concepts appear in both function types.

Feature	Linear f(x) = mx + b	Quadratic f(x) = ax² + bx + c
y-intercept	The value b — where the line crosses the y-axis	The value c — where the parabola crosses the y-axis
x-intercept(s)	Set f(x) = 0: one solution, x = −b/m	Set f(x) = 0: use the quadratic formula — can have 0, 1, or 2 solutions
Rate of change	Constant — the slope m is the same everywhere	Variable — the rate of change depends on where you are on the curve
Vertex / Turning point	None — a line has no turning point	The vertex at x = −b/(2a) is the maximum or minimum value
Graph shape	Straight line	Parabola (U-shape or upside-down U)

Comparison of key features for linear and quadratic functions

The parabola f(x) = x² − 4x + 3 with its key features labeled: the y-intercept at (0, 3), the x-intercepts at (1, 0) and (3, 0), the vertex (minimum) at (2, −1), and the axis of symmetry at x = 2.

The diagram above shows every feature the GED might ask about for a quadratic. The vertex at (2, −1) is the lowest point on this parabola because a is positive (the U opens upward). The axis of symmetry is the vertical line x = 2 that divides the parabola into two mirror-image halves. Notice that the two x-intercepts are equal distances from this axis: x = 1 is one unit to the left, and x = 3 is one unit to the right.

SECTION 6

Worked Example — Evaluating and Interpreting

A small business tracks its monthly profit using the function P(x) = −50x² + 600x − 1000, where x is the number of products sold (in hundreds). Find P(4) and explain what it means.

Evaluating a Quadratic in Context

1
Step 1 — Identify the function and the input

The function is P(x) = −50x² + 600x − 1000. We need to evaluate P(4), so we will substitute x = 4 everywhere x appears.
2
Step 2 — Substitute x = 4

P(4) = −50(4)² + 600(4) − 1000
3
Step 3 — Apply exponents first (order of operations)

(4)² = 16, so the expression becomes: P(4) = −50(16) + 600(4) − 1000

4² = 16
4
Step 4 — Multiply

−50 × 16 = −800 and 600 × 4 = 2400. Now we have: P(4) = −800 + 2400 − 1000

−800 + 2400 − 1000
5
Step 5 — Add and subtract from left to right

−800 + 2400 = 1600, then 1600 − 1000 = 600.

P(4) = 600
6
Step 6 — Interpret the result in context

Since x represents hundreds of products and P(x) represents monthly profit in dollars, P(4) = 600 means that when the business sells 400 products, the monthly profit is $600. The negative coefficient on x² tells us the parabola opens downward — there is a maximum profit, and selling too many or too few products will reduce profit.

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COMMON MISTAKE

When substituting a negative number, remember to use parentheses: (−2)² = 4, but −2² = −4. On the GED, this distinction can change your entire answer. Always wrap the substituted value in parentheses.

SECTION 7

Linear vs. Quadratic — When to Use Which

One of the most important skills on the GED is recognizing whether a situation calls for a linear or quadratic model. The table below summarizes the clues you should look for, whether you are reading a word problem, examining a data table, or studying a graph.

Clue	Linear	Quadratic
Word-problem language	"per hour," "each month," "constant rate," "flat fee plus…"	"area," "projectile," "maximum height," "profit peaks at…"
Table pattern	First differences (Δy) are constant	Second differences (Δ of Δy) are constant
Graph shape	Straight line	Curved parabola
Highest power of x	x¹ (just x)	x² (x squared)
Number of x-intercepts	Exactly 1 (unless the line is horizontal)	0, 1, or 2

How to tell whether a problem is linear or quadratic

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KEY TAKEAWAY

Think of the difference like driving a car. A linear function is like cruise control — your speed stays constant, so you cover equal distances in equal times. A quadratic function is like pressing the accelerator — your speed is itself changing, so the distances keep growing (or shrinking) over time.

SECTION 8

Connecting to More Advanced Ideas

The GED tests linear and quadratic functions because they are the building blocks of more complex mathematics. If you continue into college algebra or a career that uses data, you will encounter additional function families. The table below gives you a preview of how the skills you are learning now connect to what comes next.

What you learn on the GED	Where it leads
Evaluating f(x) by substitution	Evaluating polynomial, exponential, and trigonometric functions the same way
Finding slope (rate of change)	Calculus — finding instantaneous rate of change (derivatives)
Finding the vertex (max or min)	Optimization problems in business, engineering, and science
Interpreting y-intercept in context	Statistical regression — understanding what model parameters mean in real data

How GED function skills connect to future learning

You do not need to learn any of these advanced topics for the GED. The point is that the skills you are building right now — substituting values, reading graphs, and interpreting context — are the same skills used at every level of mathematics. Mastering them here gives you a foundation that transfers directly to college courses or career training.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A function is defined as f(x) = 2x + 7. Which statement best describes the graph of this function?

A. A U-shaped curve that opens upward
B. A straight line with a slope of 2 and a y-intercept of 7
C. A straight line with a slope of 7 and a y-intercept of 2
D. A U-shaped curve that opens downward

Show answer (B)

The correct answer is B. The function f(x) = 2x + 7 is in slope-intercept form f(x) = mx + b, where m = 2 (the slope) and b = 7 (the y-intercept). Because the highest power of x is 1, the graph is a straight line. Choice A is wrong because a U-shaped curve is a parabola, which requires an x² term — this function has no x² term, so it cannot be a parabola. Choice C reverses the slope and y-intercept — remember, in f(x) = mx + b the coefficient of x is always the slope and the constant term is the y-intercept, so the slope is 2 (not 7) and the y-intercept is 7 (not 2). Choice D is wrong for the same reason as A — a downward-opening parabola also requires an x² term (with a negative coefficient), which this linear function does not have.

Question 2: BASIC CALCULATION

A gym charges a monthly fee described by the function C(m) = 25m + 40, where m is the number of months and 40 represents a one-time signup fee. What is the total cost for 6 months of membership?

A. $150
B. $190
C. $210
D. $65

Show answer (B)

The correct answer is B. Substitute m = 6 into the function: C(6) = 25(6) + 40 = 150 + 40 = 190. The total cost for 6 months is $190. Choice A ($150) comes from computing 25 × 6 = 150 but forgetting to add the $40 signup fee — this is a common mistake of ignoring the constant term. Choice C ($210) likely results from a multiplication error, such as computing 25 × 7 + 35 or misreading the problem. Choice D ($65) appears to come from simply adding 25 + 40 = 65, treating the expression as addition of the two constants without substituting the value of m.

Question 3: INTERMEDIATE

A ball is thrown upward from a rooftop. Its height in feet after t seconds is modeled by h(t) = −16t² + 48t + 64. What is the maximum height the ball reaches?

A. 64 feet
B. 96 feet
C. 100 feet
D. 112 feet

Show answer (C)

The correct answer is C. Since a = −16 is negative, the parabola opens downward and the vertex represents the maximum height. Find the vertex using t = −b/(2a) = −48/(2 × −16) = −48/(−32) = 1.5 seconds. Now evaluate h(1.5): h(1.5) = −16(1.5)² + 48(1.5) + 64 = −16(2.25) + 72 + 64 = −36 + 72 + 64 = 100 feet. Choice A (64 feet) is the y-intercept h(0) = 64, which is merely the starting height at t = 0, not the maximum. Choice B (96 feet) results from evaluating at t = 1 instead of t = 1.5: h(1) = −16(1) + 48(1) + 64 = −16 + 48 + 64 = 96, which is not the vertex. Choice D (112 feet) likely results from an arithmetic error during substitution, such as incorrectly computing −16(2.25) as −20 instead of −36.

Question 4: APPLIED

2 pts

A city planner compares two population projections for a small town. Model A is P(t) = 500t + 12,000 and Model B is P(t) = 50t² + 12,000, where t is the number of years from now. After how many whole years will Model B first predict a larger population than Model A? Enter your answer below as a whole number of years.

Show answer

Correct answer: 11 Rubric (2 points): Point 1 — Set up and solve the inequality (1 point): Set Model B > Model A: 50t² + 12,000 > 500t + 12,000. Subtract 12,000 from both sides: 50t² > 500t. Divide both sides by 50: t² > 10t. Rearrange: t² − 10t > 0, which factors to t(t − 10) > 0. For positive values of t, this inequality is satisfied when t > 10. Point 2 — Identify the correct whole number answer with reasoning (1 point): The inequality t > 10 tells us that Model B exceeds Model A for any time after exactly 10 years. However, the question asks for a whole number of years, so we need the first whole number greater than 10, which is t = 11. We can verify: at t = 10, Model A gives 500(10) + 12,000 = 17,000 and Model B gives 50(100) + 12,000 = 17,000 — they are equal, so Model B is NOT yet larger. At t = 11, Model A gives 500(11) + 12,000 = 17,500 and Model B gives 50(121) + 12,000 = 6,050 + 12,000 = 18,050. Since 18,050 > 17,500, Model B first predicts a larger population at t = 11 years.

Question 5: CRITICAL THINKING

A company's revenue function is R(x) = −2x² + 120x and its cost function is C(x) = 20x + 400, where x is the number of units sold. Profit is defined as P(x) = R(x) − C(x). Which of the following correctly describes the maximum profit and the number of units that produces it?

A. Maximum profit of $1,050 at 25 units
B. Maximum profit of $850 at 25 units
C. Maximum profit of $1,400 at 30 units
D. Maximum profit of $800 at 20 units

Show answer (B)

The correct answer is B. First build the profit function: P(x) = R(x) − C(x) = (−2x² + 120x) − (20x + 400) = −2x² + 120x − 20x − 400 = −2x² + 100x − 400. Since a = −2 (negative), the parabola opens downward and the vertex is a maximum. Find the vertex: x = −b/(2a) = −100/(2 × −2) = −100/(−4) = 25 units. Evaluate P(25): P(25) = −2(625) + 100(25) − 400 = −1,250 + 2,500 − 400 = 850. So the maximum profit is $850 at 25 units. Choice A ($1,050 at 25 units) uses the correct x-value of 25 but forgot to subtract the $400 fixed cost, computing −1,250 + 2,500 = 1,250 − 200 or making a similar arithmetic error that omits the constant. Choice C ($1,400 at 30 units) comes from finding the vertex of the revenue function R(x) instead of the profit function: the revenue vertex is at x = −120/(2 × −2) = 30, and evaluating at x = 30 in the profit function gives P(30) = −2(900) + 100(30) − 400 = −1,800 + 3,000 − 400 = 800, not $1,400 — the $1,400 likely comes from maximizing revenue R(30) = −2(900) + 120(30) = −1,800 + 3,600 = 1,800 and then subtracting only the variable cost. Choice D ($800 at 20 units) uses an incorrect vertex at x = 20 and evaluates P(20) = −2(400) + 100(20) − 400 = −800 + 2,000 − 400 = 800; while the arithmetic is correct for x = 20, this is not the maximum since P(25) = 850 > 800.

SUMMARY

Lesson Summary

Summary

A function assigns exactly one output to each input. Linear functions have the form f(x) = mx + b and produce straight-line graphs with a constant slope (rate of change). Quadratic functions have the form f(x) = ax² + bx + c and produce parabolas whose rate of change varies. To evaluate either type, substitute the given x-value and simplify using order of operations.

Key features to identify include the y-intercept (the output when x = 0), x-intercepts (where the graph crosses the x-axis), and for quadratics, the vertex (the maximum or minimum point, found at x = −b/(2a)). To interpret a function means explaining what each part — slope, intercept, vertex — means in a real-world context. Remember: the GED formula sheet provides all the formulas you need, so focus on knowing when and how to apply them.