Interpret and Evaluate Linear and Quadratic Functions |...

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.

✦ 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.

📋 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.

Comparison of key features for linear and quadratic functions

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)

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

Step 1 — Identify the function and the inputThe function is P(x) = −50x² + 600x − 1000. We need to evaluate P(4), so we will substitute x = 4 everywhere x appears.

Step 2 — Substitute x = 4P(4) = −50(4)² + 600(4) − 1000

Step 3 — Apply exponents first (order of operations)(4)² = 16, so the expression becomes: P(4) = −50(16) + 600(4) − 1000

4² = 16

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

−800 + 2400 − 1000

Step 5 — Add and subtract from left to right−800 + 2400 = 1600, then 1600 − 1000 = 600.

P(4) = 600

Step 6 — Interpret the result in contextSince 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.

⚠️ 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.

How to tell whether a problem is linear or quadratic

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

✦ 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.

How GED function skills connect to future learning

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

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

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

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 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?

PROBLEM 4 — APPLIED

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.

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?

SUMMARY

Lesson 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.