A drone films a high school football game. The control center records its height in feet at every second: $h(t) = -5t^2 + 20t + 2$. That expression is a polynomial in $t$, and the curve that describes the flight is the graph of a polynomial function. In this section you will go beyond lines and parabolas. You will study domain and range, what happens when $x$ grows without bound, symmetry, and how zeros tell you where a function is positive or negative. Then you will use all of it to find equations and solve real problems. At the end, we will answer the question: how high does the drone get?
What is a polynomial function?
Each degree has a name: degree 0 is constant, degree 1 is linear, degree 2 is quadratic, degree 3 is cubic, and degree 4 is quartic. The domain is always $(-\infty, \infty)$, but the range depends on the type of function.
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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 $t = 3$.
State whether the parabola given by the function $g(x) = -3x^2 + 6x - 1$ opens downward or opens upward and calculate the exact coordinates of its vertex.
State whether the parabola given by the function $g(x) = -3x^2 + 6x - 1$ opens downward or opens upward and calculate the exact coordinates of its vertex.
What happens to $f(x)$ as $x$ gets huge in either direction? That is the end behavior. The degree and the sign of the leading coefficient decide it.
In limit notation, $\displaystyle\lim_{x\to +\infty} f(x) = -\infty$ means that $f(x)$ decreases without bound as $x$ grows to the right.
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Analyze the end behavior of the polynomial function $f(x) = -2x^4 + 5x^3 - x + 7$, determining the value of its limits as $x \to +\infty$ and as $x \to -\infty$.
Analyze the end behavior 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 analytical expression…
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 analytical expression and calculate all its points of intersection with the coordinate axes.
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.
Sometimes you are given points on a graph and need the equation. Choose the form that fits the information.
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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 $y = ax^2 + bx + c$.
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 $(-1, 6)$.
The factored form works for any degree: if you know all the zeros, you know the factors. But the zeros alone do not fix the polynomial.
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Find the analytical 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 analytical 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 $(0, 16)$.
Many problems ask for the values of a parameter ($k$, $m$, $b$) that make a graph behave in a certain way. The tool is almost always the discriminant of a quadratic equation.
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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.
Consider the family of quadratic functions of the form $f(x) = x^2 + bx + c$. Knowing that the graph passes through the origin and that its vertex lies on the line…
Consider the family of quadratic functions of the form $f(x) = x^2 + bx + c$. Knowing that the graph passes through the origin and that its vertex lies on the line $y = -x + 1$, find the algebraic expression for the function or functions that satisfy these conditions.
When a quantity is modeled by a downward-opening parabola, its largest value is at the vertex. For an upward-opening parabola, its smallest value is at the vertex.
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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.
A projectile is launched upwards 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 upwards 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.
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.
Back to the start: For the drone, $h(t) = -5t^2 + 20t + 2$. The vertex is at $t = -\dfrac{20}{2(-5)} = 2$ s, and the maximum height is $h(2) = -5(4) + 20(2) + 2 = 22$ ft.
Modeling with polynomials
Real problems often combine several ideas: build a function from a description, find its domain in context, then evaluate or optimize.
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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.
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.
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.