AP Precalculus: Quadratic Functions

What a Quadratic Function Is

A quadratic function is a polynomial function of degree 2. Its highest power of \(x\) is \(x^2\), and the coefficient of \(x^2\) cannot be zero. The general standard form is:

The graph of a quadratic function is a parabola. The sign of \(a\) controls whether the parabola opens upward or downward. If \(a\gt 0\), the parabola opens upward and the vertex is a minimum. If \(a\lt 0\), the parabola opens downward and the vertex is a maximum. The magnitude \(|a|\) affects vertical stretch or compression relative to the parent function \(f(x)=x^2\).

A quadratic function has domain all real numbers unless a context restricts the inputs. The range depends on the vertex and opening direction. If the vertex is \((h,k)\) and the parabola opens upward, the range is \([k,\infty)\). If the parabola opens downward, the range is \((-\infty,k]\). In AP Precalculus, the context can narrow the domain. A projectile model may use \(t\ge 0\), and a business model may restrict \(x\) to nonnegative quantities or whole units.

Quadratics are useful because they combine several AP Precalculus themes in one function family. The graph has a clear turning point. The roots connect algebra to intercepts. The average rate of change changes predictably. Equivalent forms reveal different information. A single quadratic can model height, profit, area, revenue, or any situation where the rate of change itself changes at a constant rate.

Standard Form: \(f(x)=ax^2+bx+c\)

Standard form is often the form given in algebraic problems. It is especially useful for identifying the \(y\)-intercept, applying the quadratic formula, calculating the discriminant, and finding the axis of symmetry from coefficients.

The \(y\)-intercept is \(f(0)=c\), so the graph crosses the \(y\)-axis at \((0,c)\). The axis of symmetry is:

The vertex \(x\)-coordinate is the same value. Once you find \(x_v=-\dfrac{b}{2a}\), substitute it into the function to find \(y_v=f(x_v)\). This gives the vertex \((x_v,y_v)\). If \(a\gt 0\), the vertex is a minimum; if \(a\lt 0\), the vertex is a maximum.

Example from standard form

Let \(f(x)=2x^2-8x+6\). Here \(a=2\), \(b=-8\), and \(c=6\). The graph opens upward because \(a\gt 0\). The \(y\)-intercept is \((0,6)\). The axis of symmetry is:

Evaluate \(f(2)\):

The vertex is \((2,-2)\), and the range is \([-2,\infty)\). This answer combines coefficient reading, graph interpretation and range reasoning.

Factored Form: \(f(x)=a(x-r_1)(x-r_2)\)

Factored form reveals zeros. If \(f(x)=a(x-r_1)(x-r_2)\), then \(f(r_1)=0\) and \(f(r_2)=0\). Graphically, the zeros are the \(x\)-intercepts, if the roots are real. The axis of symmetry is halfway between the real roots.

Example: \(f(x)=2(x-1)(x-5)\). The roots are \(x=1\) and \(x=5\). The axis of symmetry is \(x=\dfrac{1+5}{2}=3\). The vertex lies on this axis. Evaluate \(f(3)=2(2)(-2)=-8\), so the vertex is \((3,-8)\). Since \(a=2\gt 0\), this is a minimum, and the range is \([-8,\infty)\).

If a quadratic has a repeated real root, the factored form looks like \(f(x)=a(x-r)^2\). The graph touches the \(x\)-axis at \(x=r\) instead of crossing it. The root is also the \(x\)-coordinate of the vertex. This connects factored form, vertex form and the discriminant.

Factored form is also useful for sign analysis and inequalities. If \(f(x)=a(x-r_1)(x-r_2)\), the sign of each factor changes at the roots. That helps solve \(f(x)\gt 0\), \(f(x)\lt 0\), and related nonlinear inequalities.

Rates of Change in Quadratic Functions

College Board Unit 1 emphasizes how quantities change with respect to each other. Quadratic functions are the first major family where the average rate of change is not constant. For a linear function, equal input steps produce equal output differences. For a quadratic function, equal input steps produce first differences that change at a constant rate.

Let \(f(x)=x^2\). Over \([0,1]\), the average rate of change is \(\dfrac{1-0}{1}=1\). Over \([1,2]\), it is \(\dfrac{4-1}{1}=3\). Over \([2,3]\), it is \(\dfrac{9-4}{1}=5\). The average rates \(1,3,5\) increase by 2 each time. That constant second difference is a signature of a quadratic relationship when inputs are equally spaced.

For \(f(x)=ax^2+bx+c\) with input steps of 1, the constant second difference is \(2a\). This connects the coefficient \(a\) to the graph's concavity and rate-of-change pattern. If \(a\gt 0\), the first differences increase; if \(a\lt 0\), the first differences decrease. This is AP Precalculus reasoning because it connects a formula, a table and a graph feature.

Writing Quadratic Models from Given Features

A strong quadratic solution starts with the form that matches the data. If a problem gives a vertex, start with vertex form. If it gives roots or intercepts, start with factored form. If it gives a y-intercept or asks for coefficients, standard form may be efficient. Choosing the form is part of the mathematical reasoning, not a cosmetic step.

From a vertex and one point

Suppose a parabola has vertex \((3,-4)\) and passes through \((5,8)\). Use vertex form because the vertex is already visible:

Substitute the point \((5,8)\) to solve for \(a\):

The model is \(f(x)=3(x-3)^2-4\). This formula immediately tells you the minimum value is \(-4\), the axis of symmetry is \(x=3\), the graph opens upward and the parabola is vertically stretched compared with \(y=x^2\). If the question asks for standard form, expand only after the feature analysis is complete:

From two roots and one point

Suppose the zeros are \(-2\) and \(6\), and the graph passes through \((0,-24)\). Use factored form because the roots are already visible:

Substitute \((0,-24)\):

Now the graph features are not hidden. The \(x\)-intercepts are \((-2,0)\) and \((6,0)\). The axis of symmetry is the midpoint of the roots:

The vertex occurs at \(x=2\), and \(f(2)=2(4)(-4)=-32\). The minimum value is \(-32\). This route avoids unnecessary expansion and keeps the relationship between roots, symmetry and vertex visible.

From three points

When a problem gives three points and no clear vertex or roots, standard form is often the fastest. Let \(f(x)=ax^2+bx+c\). Suppose the points are \((0,6)\), \((1,3)\) and \((2,4)\). The first point gives \(c=6\). The second point gives:

The third point gives:

Subtract \(a+b=-3\) from \(2a+b=-1\) to get \(a=2\). Then \(2+b=-3\), so \(b=-5\). The quadratic is:

This method is also a useful check on calculator regression. A graphing calculator can fit a quadratic model to data, but if the table gives exact points that define a quadratic, you should be able to build the equation from the conditions and explain why the coefficients are determined.

From a contextual constraint

Many AP-style quadratic questions begin with a real situation instead of ordered pairs. Suppose 40 feet of fencing is used to make three sides of a rectangular pen against a wall. If \(x\) is the width perpendicular to the wall, then the length parallel to the wall is \(40-2x\). The area is:

The domain is not all real numbers. Since the width and length must be positive, \(x\gt 0\) and \(40-2x\gt 0\), so \(0\lt x\lt 20\). The vertex gives the maximum possible area:

The maximum area is 200 square feet when the width is 10 feet and the length is 20 feet. A complete AP Precalculus answer names the variables, writes the model, states the practical domain and interprets the vertex with units. That final interpretation is often what makes the difference between an algebra answer and a model answer.

Solving Quadratic Equations

Solving a quadratic equation means finding input values where the function reaches a specific output, often zero. Graphically, solving \(f(x)=0\) finds the \(x\)-intercepts. Algebraically, the best method depends on the structure of the equation.

Square-root method

Do not forget both roots when taking a square root. If \((x-4)^2=16\), then \(x-4=\pm 4\), so \(x=8\) or \(x=0\).

Factoring and zero product property

For \(x^2+5x+6=0\), factor as \((x+2)(x+3)=0\), giving \(x=-2\) and \(x=-3\).

Quadratic formula

The quadratic formula works for every quadratic equation \(ax^2+bx+c=0\), as long as \(a\ne 0\). It gives real roots when the discriminant is nonnegative and complex roots when the discriminant is negative.

Polynomial Connections: Zeros, Multiplicity and Complex Roots

A quadratic function is a second-degree polynomial function, so every quadratic idea is also part of the larger AP Precalculus Unit 1 polynomial story. The difference is that quadratics are small enough for every feature to be found exactly: vertex, intercepts, axis of symmetry, end behavior, roots and average rates of change. That makes quadratics an important training ground for later polynomial reasoning.

The Fundamental Theorem of Algebra tells you that a degree 2 polynomial has two complex zeros when zeros are counted with multiplicity. A quadratic may have two distinct real zeros, one repeated real zero or two nonreal complex zeros. The discriminant predicts which case occurs, and the graph shows how the real zeros behave.

Multiplicity is a compact way to describe how a graph behaves at an intercept. For a quadratic with two simple real zeros, the graph crosses the \(x\)-axis at each zero. For a repeated real zero, the zero is also the vertex, so the graph touches the axis and turns around. This is why \(f(x)=(x-4)^2\) has only one \(x\)-intercept but still has degree 2. The root \(x=4\) is counted twice.

Complex roots are just as important, even when a graph has no real intercepts. For \(x^2+6x+13=0\), the discriminant is \(36-52=-16\). The roots are:

The corresponding function \(f(x)=x^2+6x+13\) can also be written by completing the square:

That vertex form explains the graph. The minimum value is \(4\), so the graph stays above the \(x\)-axis. The complex roots explain the equation. Both descriptions are correct; they answer different representation questions. This is why AP Precalculus repeatedly asks students to move between graphical, numerical, analytical and verbal representations.

When not to overstate the graph

If a question asks for real zeros, a quadratic with \(D\lt 0\) has none. If a question asks for solutions over the complex numbers, it has two complex solutions. If a question asks for \(x\)-intercepts, complex solutions do not appear as points on the real coordinate plane. The wording changes the answer set, so read the question carefully before choosing a conclusion.

This also protects you from a common modeling error. A revenue model with no positive real zero in a stated domain may still be useful over that domain. A height model with one positive real zero may have another negative real zero that is algebraically valid but not meaningful for time after launch. A complete response identifies which roots belong to the model's practical domain and which roots belong only to the unrestricted equation.

For more general zero behavior beyond degree 2, use the Rational Zeros Calculator as a checking tool and read the Polynomial Functions guide for end behavior, factorization strategy and higher-degree multiplicity patterns.

Quadratic Models and Optimization

Quadratic models appear when a quantity increases and then decreases, decreases and then increases, or has a single best input. The vertex is the central model feature because it gives the maximum or minimum output. In AP Precalculus, a correct model answer should include the input value, output value and units when available.

Projectile motion

A common projectile model in feet is:

The coefficient \(-16\) comes from gravitational acceleration in feet per second squared under the standard simplified model. The graph opens downward, so the vertex gives the maximum height. If \(h(t)=-16t^2+64t+80\), then:

The maximum height is 144 feet at 2 seconds. The interpretation matters. The \(x\)-coordinate is time, and the \(y\)-coordinate is height.

Revenue model

Suppose revenue is modeled by \(R(p)=-20p^2+800p\), where \(p\) is price in dollars. The graph opens downward, so the maximum revenue occurs at the vertex:

The optimal price is 20 dollars, and the maximum revenue is \(R(20)=8000\). A complete answer should say "maximum revenue is 8000 dollars at a price of 20 dollars," if those units match the problem. Do not report only the vertex as \((20,8000)\) when the context needs a sentence.

Model limitations

College Board explicitly includes assumptions and limitations of function models in Unit 1. Quadratic models can be useful but limited. A projectile model may ignore air resistance. A revenue model may be valid only over a realistic price interval. An area model may require positive dimensions. Always check whether the context restricts the domain. The algebraic domain of a quadratic is all real numbers, but the model domain may not be.

Worked AP-Style Quadratic Examples

Example 1: Interpret all key features

Let \(f(x)=-2(x-3)^2+18\). Identify the vertex, axis of symmetry, direction, range and zeros.

The function is in vertex form. The vertex is \((3,18)\). The axis of symmetry is \(x=3\). The graph opens downward because \(a=-2\). The range is \((-\infty,18]\).

To find zeros, solve:

The roots are \(x=0\) and \(x=6\). The graph crosses the \(x\)-axis at \((0,0)\) and \((6,0)\).

Example 2: Use the discriminant before solving

Determine the number and type of roots for \(3x^2-2x+5=0\).

Since \(D\lt 0\), there are two complex conjugate roots and no real \(x\)-intercepts. The graph opens upward because \(a=3\), and it stays above the \(x\)-axis.

Example 3: Build a quadratic from roots and a point

A quadratic has roots \(x=-1\) and \(x=4\), and it passes through \((0,-8)\). Write a formula.

Start with factored form:

Use the point \((0,-8)\):

So \(f(x)=2(x+1)(x-4)\). Standard form is \(f(x)=2x^2-6x-8\).

Example 4: Use second differences

A table with equally spaced inputs has outputs \(4,9,18,31,48\). The first differences are \(5,9,13,17\). The second differences are \(4,4,4\), so the data are consistent with a quadratic model. If the input spacing is 1, then \(2a=4\), so \(a=2\). Additional information would determine \(b\) and \(c\).

Common Quadratic Function Mistakes

Official Sources Used

The AP alignment statements in this guide were checked against official College Board and AP Central sources available on July 8, 2026. The quadratic formulas are standard mathematics; the course-placement and unit-weighting statements come from official AP sources.