Polynomial Functions
A camera drone takes off to film a football game, and the control app records its height in feet every second: $h(t) = -5t^2 + 20t + 2$. That expression is a polynomial in $t$, so the graph of the flight is the graph of a polynomial function. How high does the drone get, and when? By the end of this page you will be able to answer that, and a lot more: end behavior, zeros, sign, intersections, and real-world optimization.
What Is a Polynomial Function?
Each degree has a name: degree 0 is constant, 1 is linear, 2 is quadratic, 3 is cubic, and 4 is quartic. The range depends on the family, and it is not always all real numbers.
To evaluate a polynomial, plug the input in and follow the order of operations. To find an unknown coefficient, plug in a point that is on the graph and solve for the unknown.
Calculate the value of the parameter $c$ so that the quadratic function $f(x) = 4x^2 - 3x + c$ passes through the point $(-1, 10)$.
Calculate the value of the parameter $c$ so that the quadratic function $f(x) = 4x^2 - 3x + c$ passes through the point $(-1, 10)$.
The height in feet of an observation drone as a function of time in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Calculate the height of the drone at the…
The height in feet of an observation drone as a function of time in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Calculate the height of the drone at the instants $t = 1$ and $t = 3$.
Linear Functions
Find the analytic expression of the linear function that passes through the points $A(2, -3)$ and $B(5, 6)$.
Find the analytic expression of the linear function that passes through the points $A(2, -3)$ and $B(5, 6)$.
Quadratic Functions: Vertex and Axis of Symmetry
Some problems say "concave" for a parabola that opens downward and "convex" for one that opens upward. In the US we say concave down and concave up.
Determine the coordinates of the vertex and the equation of the axis of symmetry of the quadratic function $f(x) = 2x^2 - 8x + 5$.
Determine the coordinates of the vertex and the equation of the axis of symmetry of the quadratic function $f(x) = 2x^2 - 8x + 5$.
State whether the parabola given by the function $g(x) = -3x^2 + 6x - 1$ is concave (branches downwards) or convex (branches upwards) and calculate the exact coordinates…
State whether the parabola given by the function $g(x) = -3x^2 + 6x - 1$ is concave (branches downwards) or convex (branches upwards) and calculate the exact coordinates of its vertex.
Given the quadratic function $f(x) = \frac{1}{2}x^2 - 4x + 1$, determine with reasoning its intervals of increase and decrease.
Given the quadratic function $f(x) = \frac{1}{2}x^2 - 4x + 1$, determine with reasoning its intervals of increase and decrease.
Vertex Form: Writing a Parabola from Its Vertex
If you know the vertex, vertex form is the quickest way to write the equation.
Find the equation of the parabola whose vertex is the point $V(3, -2)$ and which passes through the point $P(5, 6)$. Express the result in the form $y = ax^2 + bx + c$.
Find the equation of the parabola whose vertex is the point $V(3, -2)$ and which passes through the point $P(5, 6)$. Express the result in the form $y = ax^2 + bx + c$.
The main cable of a suspension bridge takes the shape of a parabola. The horizontal distance between the support towers is $120$ feet and the lowest point of the cable…
The main cable of a suspension bridge takes the shape of a parabola. The horizontal distance between the support towers is $120$ feet and the lowest point of the cable is $20$ feet below the attachment point on the towers. Taking the lowest point of the cable as the origin of coordinates, find the equation of the parabola and calculate the height of the cable at $30$ feet from the center.
Finding a Quadratic from Conditions
A quadratic has three unknowns, $a$, $b$, and $c$. So you need three conditions, and each one gives an equation. A point on the graph gives an equation by plugging it in. A statement about the vertex gives one using $x_v = -\frac{b}{2a}$ and $y_v = f(x_v)$.
If the equation for the unknown ends up with a negative discriminant, then no function satisfies all the conditions. That is a legitimate answer.
Determine the coefficients $a$, $b$ and $c$ of the quadratic polynomial function $f(x) = ax^2 + bx + c$ knowing that its graph passes through the points $(0, 3)$,…
Determine the coefficients $a$, $b$ and $c$ of the quadratic polynomial function $f(x) = ax^2 + bx + c$ knowing that its graph passes through the points $(0, 3)$, $(1, 4)$ and $(-1, 6)$.
Consider the family of quadratic functions of the form $f(x) = x^2 + bx + c$. Knowing that the graph passes through the origin of coordinates and that its vertex lies on…
Consider the family of quadratic functions of the form $f(x) = x^2 + bx + c$. Knowing that the graph passes through the origin of coordinates and that its vertex lies on the line $y = -x + 1$, find the analytic expression of the function or functions that satisfy these conditions.
End Behavior and Symmetry
The degree and the sign of the leading coefficient decide what happens to $f(x)$ as $x \to +\infty$ and as $x \to -\infty$. This is the end behavior of the graph.
Analyse the behavior at infinity (infinite branches) of the polynomial function $f(x) = -2x^4 + 5x^3 - x + 7$, determining the value of its limits as $x \to +\infty$ and…
Analyse the behavior at infinity (infinite branches) of the polynomial function $f(x) = -2x^4 + 5x^3 - x + 7$, determining the value of its limits as $x \to +\infty$ and as $x \to -\infty$.
A cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ is odd. Knowing that it passes through the points $(1, -4)$ and $(2, 4)$, determine its analytic expression and…
A cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ is odd. Knowing that it passes through the points $(1, -4)$ and $(2, 4)$, determine its analytic expression and calculate all its points of intersection with the coordinate axes.
Zeros and Factoring
To find zeros, set $f(x) = 0$ and factor. Take out the GCF first, then factor what is left, or use the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ when it does not factor nicely. The y-intercept is always $f(0)$.
When a cubic has no common factor and is not a difference of squares, guess a zero (try small integers like $\pm1, \pm2, \pm3$) and then use synthetic division to divide it out. The remainder is $f(\text{guess})$, so a remainder of $0$ confirms a zero, and the quotient is a quadratic you can factor. The Algebra 1 page linked at the top reviews this procedure.
Find the roots of the quadratic polynomial function $f(x) = x^2 - 5x + 6$.
Find the roots of the quadratic polynomial function $f(x) = x^2 - 5x + 6$.
Calculate analytically the points of intersection with the coordinate axes of the polynomial function $p(x) = x^3 - 4x$.
Calculate analytically the points of intersection with the coordinate axes of the polynomial function $p(x) = x^3 - 4x$.
Find the analytic expression of a cubic polynomial $P(x)$ whose only roots are $x = -2$, $x = 1$ and $x = 4$, knowing also that its graph intersects the y-axis at the…
Find the analytic expression of a cubic polynomial $P(x)$ whose only roots are $x = -2$, $x = 1$ and $x = 4$, knowing also that its graph intersects the y-axis at the point $(0, 16)$.
Sign of a Polynomial
Between two consecutive zeros, a polynomial cannot change sign, because to do that it would have to pass through $0$. So the zeros cut the number line into intervals, and you test one point in each.
Find the roots of the polynomial function $f(x) = x^3 - 6x^2 + 9x$ and study its sign over the entire real line, indicating the intervals where the function is positive…
Find the roots of the polynomial function $f(x) = x^3 - 6x^2 + 9x$ and study its sign over the entire real line, indicating the intervals where the function is positive and negative.
Intersections, Tangent Lines, and the Discriminant
The discriminant idea also counts zeros of polynomials that are quadratic in form, like $x^4 - 6x^2 + k$. Substitute $u = x^2$ to get $u^2 - 6u + k = 0$. Then remember that each positive $u$ gives two values of $x$ ($\pm\sqrt{u}$), $u = 0$ gives one, and a negative $u$ gives none.
Calculate analytically the points of intersection between the parabola $y = x^2 - 4x + 3$ and the line $y = 2x - 5$.
Calculate analytically the points of intersection between the parabola $y = x^2 - 4x + 3$ and the line $y = 2x - 5$.
Determine the values of the parameter $m$ for which the line with equation $y = mx - 4$ is tangent to the parabola $y = x^2 - 3x + 5$.
Determine the values of the parameter $m$ for which the line with equation $y = mx - 4$ is tangent to the parabola $y = x^2 - 3x + 5$.
Calculate the set of values of the parameter $k$ so that the polynomial function $P(x) = x^4 - 2x^2 + k$ has exactly two distinct real roots.
Calculate the set of values of the parameter $k$ so that the polynomial function $P(x) = x^4 - 2x^2 + k$ has exactly two distinct real roots.
Applications: Optimization
When a quantity is modeled by a downward-opening parabola, its vertex gives the maximum. For an upward-opening one, the vertex gives the minimum. The x-coordinate of the vertex tells you when or where the extreme occurs, and the y-coordinate tells you how big or small it is.
The same strategy works for every optimization problem: use the given constraint to write the quantity as a function of one variable, make sure it is quadratic, and find the vertex. For money problems, revenue is price times quantity: $R(x) = x \cdot p(x)$. For example, if the price is $p(x) = 50 - x$ dollars when $x$ units are demanded, then $R(x) = x(50 - x) = -x^2 + 50x$, with vertex at $x = 25$ and $R(25) = \$625$.
Back to the drone: for $h(t) = -5t^2 + 20t + 2$, the vertex is at $t_v = -\dfrac{20}{2(-5)} = 2$ s, and the maximum height is $h(2) = -5(4) + 40 + 2 = 22$ ft.
A projectile is launched upward so that its height in feet with respect to time in seconds follows the function $h(t) = -4.9t^2 + 19.6t + 1.5$. Calculate the maximum…
A projectile is launched upward so that its height in feet with respect to time in seconds follows the function $h(t) = -4.9t^2 + 19.6t + 1.5$. Calculate the maximum height reached by the projectile and the instant of time at which it occurs.
A farmer has $100$ feet of fencing to enclose a rectangular vegetable patch using the straight bank of a river, so no fencing is needed on that side. Express the area of…
A farmer has $100$ feet of fencing to enclose a rectangular vegetable patch using the straight bank of a river, so no fencing is needed on that side. Express the area of the patch in terms of the width $x$ perpendicular to the river and determine the dimensions that maximize this area.
The selling price in dollars of an item as a function of the number of units $x$ demanded in the market is given by the function $p(x) = 120 - 2x$. Write the total…
The selling price in dollars of an item as a function of the number of units $x$ demanded in the market is given by the function $p(x) = 120 - 2x$. Write the total revenue function $R(x)$ and calculate how many units must be sold to maximize the revenue.
Modeling with Polynomials: Profit and Volume
Polynomials of degree higher than 2 also model geometry. Here, the model alone is not the whole story: in context, the domain is limited by what makes sense physically.
A technology company sells $x$ units of a certain electronic component. The unit selling price modeled in dollars is $p(x) = 200 - 0.5x$. The total production costs…
A technology company sells $x$ units of a certain electronic component. The unit selling price modeled in dollars is $p(x) = 200 - 0.5x$. The total production costs follow the function $C(x) = 5000 + 20x$. Express the profit function $B(x)$, determine the number of units that maximizes the profit and calculate the amount of this maximum profit.
A rectangular sheet of cardboard $40$ in. long by $30$ in. wide is available. We want to build a box without a lid by cutting out a square of side $x$ from each of the…
A rectangular sheet of cardboard $40$ in. long by $30$ in. wide is available. We want to build a box without a lid by cutting out a square of side $x$ from each of the corners and folding up the resulting flaps. Express the volume $V(x)$ of the box as a polynomial function, state its valid domain in the context of this geometric problem and calculate the volume if a square of side $5$ in. is cut out.