Difficulty: Beginner to Intermediate | Prerequisites: Exponents, basic function notation, plotting points on the coordinate plane.
Tags: graphing polynomials, polynomial degree, end behaviour, leading coefficient, zeros of a polynomial, x-intercepts, turning points, matching equations to graphs, polynomial functions, linear vs cubic vs quartic
Graphing polynomials is about connecting the algebra (the equation) to the geometry (the shape on the coordinate plane). Once you know a few key properties of a polynomial, such as its degree, its leading coefficient, and its zeros, you can predict what the graph looks like without plotting dozens of points. This topic builds directly on your work with exponents and basic function evaluation, and it sets the stage for later work on factoring, the Remainder Theorem, and polynomial regression.
Every polynomial's shape is controlled by its degree (how many direction changes it can have) and its leading coefficient (which way the ends of the graph point). Find the degree, check the sign of the leading coefficient, locate the zeros, and you can sketch or identify the graph.
Polynomial function
A function built from terms of the form axⁿ, where n is a non-negative integer. Examples include y = 7x + 5 and y = x⁴ − 3x³ − 2x² + 4x. In simple terms, it is any expression made of variables raised to whole-number powers, multiplied by coefficients, and added together.
Degree of a polynomial
The highest power of x that appears in the polynomial. For y = −2x³ − 3x² + 7x + 5, the degree is 3. Think of it as the single number that tells you the most about the polynomial's overall shape and behaviour.
Leading coefficient
The coefficient attached to the highest-degree term. In y = −2x³ − 3x² + 7x + 5, the leading coefficient is −2. In simple terms, this number (and especially its sign) controls whether the graph's ends point upward or downward.
End behaviour
What happens to the y-values as x moves toward positive or negative infinity. Think of it as the answer to "which direction do the two tails of the graph point?"
Zeros (roots, x-intercepts)
The x-values where the polynomial equals zero, i.e. where the graph crosses or touches the x-axis. If you can factor the polynomial, each factor gives you a zero.
Turning point
A point where the graph changes from increasing to decreasing, or vice versa. A polynomial of degree n can have at most n − 1 turning points.
Linear polynomial
A degree-1 polynomial. Its graph is always a straight line. Example: y = 7x + 5.
Cubic polynomial
A degree-3 polynomial. Its graph can have up to 2 turning points and always has opposite end behaviour on the left and right. Example: y = −2x³ − 3x² + 7x + 5.
Quartic polynomial
A degree-4 polynomial. Its graph can have up to 3 turning points, and both ends point the same direction. Example: y = x⁴ − 3x³ − 2x² + 4x.
The degree tells you two things at a glance:
Even degree (2, 4, 6, ...): both ends of the graph point the same way. If the leading coefficient is positive, both ends go up. If negative, both ends go down.
Odd degree (1, 3, 5, ...): the ends point in opposite directions. Positive leading coefficient means the graph falls on the left and rises on the right. Negative leading coefficient means it rises on the left and falls on the right.
This is the fastest way to narrow down which equation matches which graph. If a graph's two tails both point upward, you are looking at an even-degree polynomial with a positive leading coefficient.
Equation 1: y = 7x + 5
Degree: 1 (linear)
Leading coefficient: +7
Graph: a straight line with slope 7 and y-intercept 5
End behaviour: falls to the left, rises to the right (odd degree, positive leading coefficient)
Zeros: set 7x + 5 = 0, giving x = −5/7 ≈ −0.71
Turning points: none (a line never changes direction)
Equation 2: y = −2x³ − 3x² + 7x + 5
Degree: 3 (cubic)
Leading coefficient: −2
End behaviour: rises on the left, falls on the right (odd degree, negative leading coefficient)
Maximum turning points: 2
y-intercept: plug in x = 0, get y = 5
To find the zeros exactly you would factor or use a graphing tool; the important skill here is recognising the S-shaped cubic curve with its ends going in opposite directions
Equation 3: y = x⁴ − 3x³ − 2x² + 4x
Degree: 4 (quartic)
Leading coefficient: +1
End behaviour: both ends rise (even degree, positive leading coefficient)
Maximum turning points: 3
y-intercept: y = 0 (the constant term is zero)
You can factor out x straight away: y = x(x³ − 3x² − 2x + 4), so x = 0 is one zero
A reliable process:
Check the ends of the graph first. Both pointing up or both pointing down means even degree. Opposite directions means odd degree.
Check the sign: if both ends go up, the leading coefficient is positive. If both go down, it is negative.
Count the turning points. A graph with 2 turning points is at least degree 3. A graph with 3 turning points is at least degree 4.
Check the y-intercept by reading where the graph crosses the vertical axis, then compare with the constant term in the equation (the value you get when x = 0).
Check x-intercepts if visible. If the graph crosses the x-axis at a point you can read off, substitute that x into the candidate equation and see whether y = 0.
A degree-n polynomial has at most n − 1 turning points (it can have fewer).
A degree-n polynomial has at most n real zeros (it can have fewer).
A linear polynomial (degree 1) has exactly 1 zero and 0 turning points.
A cubic (degree 3) has at most 3 real zeros and at most 2 turning points.
A quartic (degree 4) has at most 4 real zeros and at most 3 turning points.
End behaviour summary (the cheat sheet):
Degree | Leading coefficient | Left end (x → −∞) | Right end (x → +∞) |
|---|---|---|---|
Odd | Positive (+) | Falls (y → −∞) | Rises (y → +∞) |
Odd | Negative (−) | Rises (y → +∞) | Falls (y → −∞) |
Even | Positive (+) | Rises (y → +∞) | Rises (y → +∞) |
Even | Negative (−) | Falls (y → −∞) | Falls (y → −∞) |
Maximum turning points = degree − 1
Maximum real zeros = degree
Polynomial models show up whenever something accelerates, decelerates, or changes direction. Engineers use cubic and quartic polynomials to model the deflection of beams under load, and economists use them to approximate cost curves that have diminishing and then increasing marginal costs. Even the trajectory of a ball (a degree-2 polynomial in time) is a special case of this same toolkit.
Students often assume that a polynomial of degree n must have exactly n real zeros. It does not. It can have fewer, because some zeros may be complex (non-real) or repeated.
Students sometimes confuse the degree of the polynomial with the number of terms. A polynomial like y = x⁴ has degree 4 but only one term. Degree is about the highest exponent, not the number of pieces.
A common mistake is reading end behaviour backwards: thinking a negative leading coefficient on an odd-degree polynomial means "falls left, rises right." It is the opposite. Negative leading coefficient on odd degree means "rises left, falls right."
Students sometimes forget that the y-intercept is just the constant term. For y = x⁴ − 3x³ − 2x² + 4x, the constant term is 0, so the graph passes through the origin.
⚠️ End behaviour questions are among the most common multiple-choice items on polynomial tests. Know the table above cold.
⚠️ "How many turning points can this polynomial have?" is a staple exam question. The answer is always "at most degree minus one."
⚠️ Matching equations to graphs is a frequent assessment format. The fastest approach: check ends first, then count turning points, then verify with the y-intercept.
⚠️ If you can factor out an x from every term (as in Equation 3), say so. Examiners reward recognising that x = 0 is a zero.
True or false: A degree-4 polynomial always has 4 x-intercepts. False. It has at most 4, but some zeros may be complex.
True or false: If both ends of a graph point downward, the leading coefficient is negative. True.
Fill in the blank: A polynomial of degree 5 can have at most ___ turning points. 4.
True or false: The graph of y = 7x + 5 has one turning point. False. A linear function has zero turning points.
Fill in the blank: The y-intercept of y = x⁴ − 3x³ − 2x² + 4x is ___. 0.
Q: What is the degree and leading coefficient of y = −2x³ − 3x² + 7x + 5?
A: Degree 3, leading coefficient −2.
Q: Describe the end behaviour of y = −2x³ − 3x² + 7x + 5.
A: Because the degree is odd and the leading coefficient is negative, the graph rises on the left (as x → −∞, y → +∞) and falls on the right (as x → +∞, y → −∞).
Q: How many turning points can y = x⁴ − 3x³ − 2x² + 4x have at most?
A: 3 (degree 4 minus 1).
Q: You see a graph where both tails point upward and there are 3 turning points. Which of the three equations from this worksheet could it match?
A: Equation 3, y = x⁴ − 3x³ − 2x² + 4x. It is degree 4 (even) with a positive leading coefficient, so both ends rise, and it can have up to 3 turning points.
Q: What is the y-intercept of y = 7x + 5, and what does it tell you about the graph?
A: The y-intercept is 5. The graph crosses the vertical axis at (0, 5).
Q: Factor out the greatest common factor from y = x⁴ − 3x³ − 2x² + 4x. What zero does this immediately give you?
A: y = x(x³ − 3x² − 2x + 4). The factor x tells you that x = 0 is a zero, meaning the graph passes through the origin.
This material connects directly to factoring polynomials: once you can identify zeros from a graph, you can write the polynomial in factored form, and vice versa. It also sets up the Rational Root Theorem and synthetic division, which are techniques for finding exact zeros of higher-degree polynomials when a graph alone is not precise enough. If you go on to calculus, derivatives give you a way to find turning points algebraically rather than reading them off a graph.
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