Functions and Graphing
5 Problems • 8 sub-topics
Adalynn Le • 5/6/2026
Table of Contents
Introduction
Functions and graphing go hand in hand when it comes to modeling and visualizing problems. Almost all equations can be graphed. Functions particularly refer to when each output has exactly one input, and graphing refers to the visual representation of such equations.
Linear Functions
We've all seen this, a straight line modeled by \(y=mx+b\). That being said, it's a great warmup function to learn about, and I think that through these excercsies you'll find it's a lot more complex than you may think.
Equation
\(y=mx+b\)
\(x=\) input
\(y=f(x)\) (the output)
\(m=\) slope, calculated as the change in \(x\) over the change in \(f(x)\)
\(b=y-\)intercept, or the value that \(f(x)\) is at for \(x=0\)
Drag and Drop
\(=\)\(+\)Graph
Slope:
Y-Intercept
Linear functions go hand in hand with a few other terms. The first is an arithmetic sequence. Just like in a linear relationship, each value differs from the prior one by a constant value, known as the common difference. In arithmetic sequences, you can even have a non-zero starting value \(a\) which is not the case for proportional relationships. Remember that although proportional relationships also have a common difference they must always start at \(0\), or pass through the \(y-\)intercept at \(0\).The reason for this is that proprotional relationships are defined as functions where each output is proportional to its input. While by nature this ensures a common difference, it is also only possible to be expresed as \(y=mx\), without \(b\) or a noticeably \(y-\)intercept
Quadratics and Polynomials
A polynomial is defined as an expression with multiple terms of varying exponential power. The most common is a quadratic, but it is important to realize that similar ideas, equations, and metrics, apply to most other polynomial functions.
Quadratics
A quadratic is a polynomial where the highest power is \(2\). These can be expressed in \(3\) main ways, with each form showing a different metric. The two biggest parts of a quadratic parabola are its roots, and its vertex. Every parabola must have a vertex, although not all have roots. The vertex is the externum of the parabola, either the minimum or maximum depending on which way the parabola is "facing" (see the grpah below). The roots are where the graph intercepts the \(x-\)axis.
Standard Form - \(ax^2+bx+c\)
The standard form of a quadratic equation is the jumping point for converting into other forms. It is particularly helpful because it allows you to easily see \(a, b, c\) which are nescessary for the quadratic equation. Standard form is written as \(ax^2+bx+c\). From here, you could factor for factored form or complete the square for vertex form. From here, you can also disect \(a, b,\) and \(c\) and input those into the equation \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\) which allows you to find all roots even if they are imaginary or irrational.
Vertex Form - \((x-h)^2+k\)
This form directly shows you the vertex, or externum for a parabola: \(k,h\). In order to find this form, you need to perform a method known as completing the square. Considering only \(ax^2+bx\), you find the nescessary value to make \(ax^2+bx+z\) a perfect square of some equation \(\sqrt{a}x^2+\frac{b}{2}\) (so essentially \(z=\frac{b}{2}^2\)). Then, in order to keep the equation equivalent, you subtract that value \(z\) from \(c\) to get \(k\).
Factored Form - \((x-z)(x-q)\)
In this example, \(z\) and \(q\) are the roots of the equation, or the values of \(x\) such that \(f(x)=0\). If you want to learn more about factoring, you can check out our article on algebraic manipulation, but in the factored form, it is nescessary to find values for \(z\) and \(q\) such that \(-z \cdot -q = c\) and \(-z + (-q)=b\). Note, that in a situation where \(a \neq 1\), \(x\) will also have coefficients that need to be factored. This is the most effective form for finding roots of a quadratic. Not all equations can be factored like this because they might have imaginary or irrational roots.
Graphing the Quadratic
The general shape of a quadratic will always be a parabola. The width, eccentricity, vertex, and roots can all be manipulated, but the shape remains the same. When graphing the parabola, it is important to make use of the vertex and roots. Often, I would choose to start with mapping those out and connecting them in the general shape of the parabola. This is great for visualizing but if you are looking for solutions or something that requires presciscion, you may need to be more exact. I also want to take this moment to mention that not all parabolas has roots. For xample, imagine a parabola with a vertex above the \(x-axis\) that is concave up ("opens" upwards), then how will it cross the \(x\) axis? It is also possible for the vertex to be on the \(x-\)axis. In order to predict this, we use something called "the discriminant", and it's what is stored inside the radical of the quadratic equation. The equation is \(b^2 - 4ac\). If the discriminant is positive, there will be two solutions. If zero, one (the vertex) and if negative, none. Since the quadratic equation returns solutions, it makes sense. If it's negative, you can't take the root of a negative, thus the solutions are not real. If zero, the square root is ONLY \(0\), there is no positive and negative variant, and so on.
Just like with any function, each value means something. Let's return to our basic standard form equation: \(ax^2+bx+c\). \(a\) is something I like to think of as "amplitude", it is the vertical stretch of a quadratic. A higher value of \(a\) stretches the parabola vertically, or squashes it horizontaly. Next is \(b\). This affects the phase shift (horizontally) BUT also the vertical displacement. Think about it, when you complete the square, the value of \(\frac{b}{2}\) affects the value inside the parentheseis for the \(x\), but also changes \(k\). Thus, there's a phase shift in both. Finally, \(c\) is just a value added to the final value \(f(x)\), thus moving it upwards
Other Polynomials
It's pretty much impossible to know everything about every polynomial, but there are a few basic rules when it comes to predicting their behavior.
End Behavior:For all polynomials where the highest power is even, the end behaviors are the same for both sides. For all polynomials where the highest power is odd, the end behavior is different on teither side. The graph will face downwards for leading coefficient \(a \le 0\)
Zeroes:Allthough something similar to a discriminant will exist for all polynomials, effectively an expression that tells you the exact number of zeroes, it often gets very complex very easily. Thus, for larger polynomials, we use Descarte's Rule of Signs-a trick made my French mathematician Descarte to narrow down the number of zeroes in a function. The rule tells you that counting the number of sign changes in each term for \(f(x)\) gives the maximum number of zeroes, and any positive number that is the maximum minus an even number is also possible. The same is true for negative roots and \(f(-x\)). The reason for this has to do with multiplicity
Multiplicity:In a factored form, the power each factor expression is raised to determines it's behavior when crossing the \(x-\)axis. For those with odd powers, known as having an odd multiplicity, they will cross through. For even, they will "bounce" off of the \(x-\)axis, essentially staying on the same side of it
Exponential Function
The exponential function is written with a real number base and \(x\), or the input, as the exponent. It models rapid and often uncontroled growth that can be used for compound interest, culture growths, etc. It's parent function is \(f(x)=2^x\), but it also exists commonly with a base of \(e\) or \(10\). Notice that with a positive base, the value will never be negative. Just as with a typical equation, you can perform operations by adding a coefficent to \(2^x\) (notice it must be \(z(2^x\))) to stretch, add a constant \(c\) to add a vertical shift, and change the value of \(x\) in the exonent for a horizontal shift \(2^{x-z}\). Notice that the exonential change is not always growth. For bases such that base \(b\) falls in \(0 \le b \le 1\), the graph will take the shape of decay. For a negative value \(b\), the behavior of the graph is od and will switch into two divergent, dottted graphs. Most modern graphing calculators cannot handle this operation, so it is not very common.
Inverse Equations
Radical Functions
An exponential function is made when the input \(x\) is nested inside a radical . The shape of it is similar to half of a sideways parabola, and that's for good reason. The radical parent function \(f(x)=\sqrt{x}\) is what we call the inverse of the parent quadratic function \(f(x)=x^2\). That is to say if you switched the positions of \(f(x)\) and \(x\) and proceeded to solve for \(f(x)\), you would end up with the other equation. It also means that what we percieved as a sideways parabola is a reflection of the other over the line \(y=x\) which is true of all inverses. Also, make sure you notice that radical functions are technically not functions because for even radicals they can return both positive and negative values. That being said, it is customary to take the positive value unless otherwise specified.
Logarithmic Function
Just a radical function is the inverse of a quadratic or polynomial, the logarithmic function is the inverse of the exponential function. You'll notice it looks quite a bit like a radical function, and that makes sense, because the exponential function looks like a polynomial function. I want to take this moment to clear the distinction. Although they look the same, they obviously aren't, because one has \(x\) as a base and the other has \(x\) as a power. What this means is that, assuming the exponent/base is \(2\), the slope of the quadratic at a given point is \(2x\) and the slope of an exponential at a given point is \(2^x \cdot \textup{ln}(a)\). This doesn't matter so much, but it is an important distinction that the two graphs are pretty different and have different shapes.
Graph of \(x^2\)
Trigonometry and Waves
You are not required to know trigonometry for the AMC 10. In my overview of the last \(300\) questions from the past \(5\) years, only \(2\) really needed trigonometry. That being said, it is important to know in general, and having a basic knowledge of periodic motion and oscillations in general. A sine or cosine looks like a wave due to the fact that sine is based on the coordinates of circular motion. This is helpful for when you have to calculate periodic motion, although, like I said, it most likely won't appear on the AMC 10. The basic formula for a sinusodial equation is \(f(x)=A\textup{sin}(Bx+C)+D\) with \(A=\)amplitude (height), \(\frac{2\pi}{B}=\)period (distance between troughs), \(C=\) phase shift horizontally, and \(D=\)vertical shift.Other trigonomectric functions such as \(\textup{tan, cos, sec}\) have vertical squiggly-style grpahs, although these have never come up on the AMC 10 before.
Reciprocal Functions
Reciprocal functions are where things get sort of blurred, because there are a lot of things to think about. Reciprocal functions occur when \(x\) is in the denominator of a fraction. The numerator can have \(x\) as well. The technical definition is that a rational function is the ratio of two polynomial equations. The shape of a polynomial graph can differ quite a bit, but it typically typically looks like an exponential decay graph and another rotated so that it is in Quadrant \(\textup{III}\)
The most important things to remember about reciprocal functions are that they will have discontinuities and asymptotes. Asymptotes are values of \(x\) or \(y\) that the graph approaches but never reaches. The vertical asymptote, the value of \(x\) that is aproached, can be found by solving for the value of \(x\) that makes the denominator equal to \(0\). This makes sense, because dividing by zero is impossible and thus undefined. Horizontal asymptotes are values of \(y\) that are approached but never fully reached. Since a rational function is written as the ratio of two polynomails, the asymptote depends on the powers of the polynomials. For \(\frac{x^n}{x^m}\) where \(n \le m\), the asymptote is \(0\), because the growth of the denominator is increasing faster than the numerator, making it get closer to \(0\). For \(n=m\), the ratio is that of the leading base of the numerator to the leading base of the denominator. Form \(n>m\), there is no asymptote.
Discontinuities are similar to asymptotes in the fact that they are values that are not defined for a rational function. What's different though is that in an asymptote, the values get consistenly closer to that value without touching. In a disconinuity, the line seemingly passes through the point. Discontuinuities are created because it is impossible to divide by \(0\). Consider the equation \(\frac{x^2}{x}\) which simplifies down to \(x\), which is just a linear graph, right? Not exactly, because when \(x=0\), the equation simplifies down to \(\frac{0}{0}\) which is undefined. Even if it may look and act the same as \(y=x\), there is a small discontinuity there that needs to be noted.
Modifiers
The two following operations are what I like to call "modifiers" because they can be applied to or in the context of another funcion. They may change the expression of a graph, but the general shape remains the same.
Absolute Value
The absolute value function is a function that makes all the values of \(f(x)\) positive. For equations symetric about the origin, it reflects Quadrant \(III\) up and over the \(x-\)axis. Essentially, anything negative is fliped over \(y=0\). This actually does come up quite a bit in the AMC 10, not so much in word problems bust just in equations where you are solving for a value or especially solving for a bounded area. Note that the absolute value can also be applied around the output \(f(x)\) to create a crosshair-style shape.
Floor Functions
This is one of the AMC 10's favorite tricks. The floor function, written as \(\lfloor{x}\rfloor\) takes the largest integer smaller than \(x\). This is important because it creates a graph that looks like "steps" with a bunch of discontinuities. Like absolute value. This doesn't come up so much in word problems or applications, instead mostly being focused in solving systems of equations or solving for areas on the coordinate grid.
Graph of \(x^2\)
Graph of \(x^2\)
Graphing Tips
Although you will not be given a calculator or graphing tool on the AMC 10 there are going to be questions where it helps you to solve and visualize a question. That being said, it is important to be careful and precise, especially when solving for intersections and solutions to systems of equations.
On the AMC 10 you are given graph paper and a ruler, which is really all you need. Even for nonlinear equations, you can plot points and draw the connections between them for good graphing
It is also essential to use apropriate scale. In m yexperience with questions you need to graph, it typically does not extend more than \(5\) units in any direction and definitely not \(10\). This could help you when drawing your coordinate grid.
Of course, you need to know what to graph. This can be done by simplly having a solid foundation and understanding of the parent functions and how they are manipulated by transformations. Recall that transformations typically follow the following rules:
- If it is a coefficient of the input \(x\), it affects the height and "stretches" the shape
- The horizontal phaseshift is writen as \((x-h)\), where it is subtracted from wherever \(x\) is. Notice that this is different from a vertical shift because it is subtracted, it is applied in \(x\) (so if there's a coefficient it distributes or if \(x\) is squared you square the whole expression), and it affects horizontal phaseshift
- The vertical shift is writen as a constant added to the equation at the end. It literally adds a value to the vertical output \(f(x)\)
Conclusion
Graping is an essential skill on the AMC 10 and will help you visualize and model systems. In order to graph well, it is nescessary to have a thorough understanding of functions and how to graph them, as well as their behaviors. This is just a quick rundown of the essential functions, but for more complex AMC 10 questions that combine or rearrange equations, I would check out our articles on algebraic manipulation and systems of equations (coming soon).
Question 1:
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