Quadratic Functions and Parabolas
Toss a ball straight up. It rises, slows down, stops for an instant at the very top, and then falls back. Suppose its height after $t$ seconds is $h(t) = -16t^2 + 32t + 4$ feet. How high does it get, and when? The top of that path is the vertex of a parabola, and you will be able to find it with one short formula.
Quadratic functions and their graphs
Two things show up right away. The number $c$ is the y-intercept, because $f(0) = c$. The sign of $a$ tells you which way the parabola opens.
Play with the sliders. Watch what $a$ does, and watch where the curve crosses the y-axis as you change $c$.
To check whether a point is on the graph, substitute its coordinates into the equation. The point is on the graph only if both sides come out equal.
Parabolas also model motion. Here the input is time and the output is a distance.
Determine, by substituting into the equation, whether the point $(-1, 4)$ and the point $(2, 1)$ belong to the graph of the parabola $y = 2x^2 - 3x - 1$.
Determine, by substituting into the equation, whether the point $(-1, 4)$ and the point $(2, 1)$ belong to the graph of the parabola $y = 2x^2 - 3x - 1$.
The function $d(t) = 5t^2$ describes the distance in inches that a marble travels when rolling down an inclined track as a function of time $t$ in seconds. What distance…
The function $d(t) = 5t^2$ describes the distance in inches that a marble travels when rolling down an inclined track as a function of time $t$ in seconds. What distance will it have traveled after $4$ seconds? How long will it take to travel exactly $125$ in?
Look at the three parabolas represented in the following figure. Match each of the analytical expressions with its corresponding graph (A, B or C). 1) $y = -x^2 + 2$ 2)…
Look at the three parabolas represented in the following figure. Match each of the analytical expressions with its corresponding graph (A, B or C).
1) $y = -x^2 + 2$
2) $y = x^2 - 4x + 4$
3) $y = x^2 - 4$
The vertex and the axis of symmetry
A parabola is a mirror image of itself across a vertical line, the axis of symmetry. The vertex is where the axis meets the curve. Let's find it by completing the square.
Doing the same steps with letters gives a formula for every parabola. Factor $a$ out of the first two terms and complete the square inside:
$$f(x)=a\left(x^2+\frac{b}{a}x\right)+c = a\left(x+\frac{b}{2a}\right)^2 + c-\frac{b^2}{4a}$$The squared part is $0$ when $x=-\dfrac{b}{2a}$.
Because of the symmetry, two points at the same height are the same distance from the axis. For example, $(0,c)$ and its mirror image are at the same height $c$, so the axis is exactly halfway between them.
Given the quadratic function $f(x) = 2x^2 - 8x + 5$, determine the exact coordinates of its vertex and the equation of its axis of symmetry.
Given the quadratic function $f(x) = 2x^2 - 8x + 5$, determine the exact coordinates of its vertex and the equation of its axis of symmetry.
State which way the parabola $y = -3x^2 + 6x - 1$ opens, justifying your answer, and find out whether the vertex of this function corresponds to an absolute maximum or…
State which way the parabola $y = -3x^2 + 6x - 1$ opens, justifying your answer, and find out whether the vertex of this function corresponds to an absolute maximum or minimum. Calculate its coordinates.
Intercepts and the discriminant
The y-intercept is always unique: $f(0)=c$. The x-intercepts (also called zeros or roots) come from solving $ax^2+bx+c=0$ by factoring or with the quadratic formula. How many solutions there are depends on the discriminant $b^2-4ac$.
Calculate the points of intersection with the coordinate axes (both the x-axis and the y-axis) of the parabola $y = -x^2 + 2x + 8$.
Calculate the points of intersection with the coordinate axes (both the x-axis and the y-axis) of the parabola $y = -x^2 + 2x + 8$.
Consider the family of parabolas given by the equation $y = x^2 - kx + 9$. Determine the values of the parameter $k$ for which the parabola is tangent to the x-axis…
Consider the family of parabolas given by the equation $y = x^2 - kx + 9$. Determine the values of the parameter $k$ for which the parabola is tangent to the x-axis (that is, it has a single point of intersection with the x-axis).
Factored form
If you know the x-intercepts $r_1$ and $r_2$, you can write the function right away.
Change $a$ in the picture: the zeros stay at $-2$ and $4$, but the parabola gets wider, narrower, or flips.
Find the analytical expression of the quadratic function $f(x) = ax^2 + bx + c$ knowing that it intersects the x-axis at the points $x = -1$ and $x = 5$, and that the…
Find the analytical expression of the quadratic function $f(x) = ax^2 + bx + c$ knowing that it intersects the x-axis at the points $x = -1$ and $x = 5$, and that the maximum value reached by the function is $18$.
Vertex form and translations
Completing the square gives a third form of the same function.
You can also go from standard form to vertex form by completing the square.
Some problems call $y=x^2$ the parent parabola. That is the starting point for every translation above.
Express the function $y = (x + 3)^2 - 2$ as the result of applying a translation to the parent parabola $y = x^2$. Clearly state the direction and the number of units…
Express the function $y = (x + 3)^2 - 2$ as the result of applying a translation to the parent parabola $y = x^2$. Clearly state the direction and the number of units shifted on both the horizontal and vertical axes.
Look at the graph of the parabola. Determine the coordinates of its vertex and use an additional point on the graph to deduce its algebraic expression in the form…
Look at the graph of the parabola. Determine the coordinates of its vertex and use an additional point on the graph to deduce its algebraic expression in the form $y = a(x-h)^2 + k$. Finally, expand it to the form $y = ax^2 + bx + c$.
Writing the equation of a parabola
Which form should you use? It depends on what you are given.
| You know | Start with |
|---|---|
| The vertex and one more point | Vertex form $a(x-h)^2+k$ |
| The x-intercepts and one more point | Factored form $a(x-r_1)(x-r_2)$ |
| Three points (or points and other conditions) | Standard form $ax^2+bx+c$ and a system of equations |
For standard form, each point $(x_0,y_0)$ gives one equation: substitute it into $y=ax^2+bx+c$. Three unknowns need three equations.
Try this in the picture: with $c=2$, find the values of $a$ and $b$ so that the curve passes through $(1,3)$ and $(2,8)$. You should land on the answer above.
A special case: if the function is $f(x)=ax^2+c$ (no $x$ term), the vertex is on the y-axis. The point with $x=0$ gives $c$ immediately, and a second point gives $a$.
Knowing that the graph of the quadratic function $f(x) = ax^2 + c$ passes through the points $(0, 3)$ and $(2, 11)$, set up a system of equations and calculate the…
Knowing that the graph of the quadratic function $f(x) = ax^2 + c$ passes through the points $(0, 3)$ and $(2, 11)$, set up a system of equations and calculate the values of the parameters $a$ and $c$.
Determine the expanded algebraic expression of a quadratic function $f(x) = ax^2 + bx + c$ knowing that its vertex is the point $V(2, -1)$ and that it passes through the…
Determine the expanded algebraic expression of a quadratic function $f(x) = ax^2 + bx + c$ knowing that its vertex is the point $V(2, -1)$ and that it passes through the point $P(4, 3)$.
Calculate the coefficients $a$, $b$ and $c$ of the quadratic function $y = ax^2 + bx + c$ knowing that its graph passes through the points $(0, -5)$, $(1, -6)$ and…
Calculate the coefficients $a$, $b$ and $c$ of the quadratic function $y = ax^2 + bx + c$ knowing that its graph passes through the points $(0, -5)$, $(1, -6)$ and $(-1, -2)$.
Modeling with parabolas
Arches, fountains and shots all follow parabolic paths. To write an equation, you choose where to put the origin. A smart choice (the vertex on the y-axis, or the start of the path at the origin) makes the numbers simple.
To prove that a path does (or doesn't) go through a given point, substitute its $x$-coordinate and compare the result with the point's $y$-coordinate.
An arch of a stone bridge has the shape of a parabola. The base of the arch (at water level) is $12$ feet wide and the maximum height in the center is $6$ feet. If we…
An arch of a stone bridge has the shape of a parabola. The base of the arch (at water level) is $12$ feet wide and the maximum height in the center is $6$ feet. If we place the origin of coordinates at the center of the base of the arch, find the equation of the parabola that describes its profile.
A basketball player shoots at the basket from a height of $2$ feet. The ball follows a parabolic trajectory, reaching its maximum height of $4$ feet at a horizontal…
A basketball player shoots at the basket from a height of $2$ feet. The ball follows a parabolic trajectory, reaching its maximum height of $4$ feet at a horizontal distance of $2$ feet from the point of release. Knowing that the hoop of the basket is at a height of $3$ feet and at a horizontal distance of $4.5$ feet, deduce the equation of the trajectory and prove analytically whether the ball will go through the hoop or not.
Maximum and minimum values
The vertex is the answer to every "greatest" or "least" question about a quadratic model. When $a<0$ you get a maximum; when $a>0$ a minimum. No graphing needed.
During a training session, a volleyball player throws the ball upwards. The height $h$ (in feet) reached by the ball as a function of time $t$ (in seconds) is given by…
During a training session, a volleyball player throws the ball upwards. The height $h$ (in feet) reached by the ball as a function of time $t$ (in seconds) is given by the expression $h(t) = -5t^2 + 10t + 2$. What is the maximum height reached by the ball and at what exact instant does it occur?
A company that manufactures personalized cell phone cases has determined that if it sells each case for $\$x…
A company that manufactures personalized cell phone cases has determined that if it sells each case for $\$x$, its daily revenue in dollars is given by the quadratic function $I(x) = -2x^2 + 48x$. At what price must they sell the cases to maximize their daily revenue? What does this maximum revenue amount to?
Area and profit problems
Many optimization problems give you a constraint (a fixed amount of fence, or a cost) and ask for the best choice. The plan:
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Name the unknown $x$ and write the other quantities in terms of $x$ using the constraint.
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Write the quantity to optimize as a quadratic function of $x$.
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Find the vertex. Check that $x_v$ makes sense (lengths must be positive).
For a rectangle with a fixed perimeter $P$ and one side $x$, the other side is $\dfrac{P}{2}-x$. If a wall replaces one side, it changes the constraint, as in the next example.
We have $40$ feet of rope to mark out a rectangular play area in a camp. If we call $x$ the length of the base of the rectangle, write the function that determines the…
We have $40$ feet of rope to mark out a rectangular play area in a camp. If we call $x$ the length of the base of the rectangle, write the function that determines the area of the enclosure as a function of $x$. What must the dimensions of the rectangle be so that the enclosed area is as large as possible?
A farmer wants to build a rectangular pen for his chickens using an existing long stone wall, so that he only needs to fence three of the four sides of the rectangle. He…
A farmer wants to build a rectangular pen for his chickens using an existing long stone wall, so that he only needs to fence three of the four sides of the rectangle. He has $100$ feet of wire fencing. Determine the dimensions of the pen so that its area is as large as possible and calculate this maximum area.
An electric scooter factory has fixed monthly expenses of $\$2000$ and producing each scooter costs an additional $\$15$. The monthly revenue obtained from the sale of…
An electric scooter factory has fixed monthly expenses of $\$2000$ and producing each scooter costs an additional $\$15$. The monthly revenue obtained from the sale of $x$ scooters is given by the function $I(x) = -0.1x^2 + 65x$. Write the function that represents the monthly profit $B(x)$. How many scooters must they manufacture and sell to obtain the maximum profit?
Intersections with lines and other parabolas
Where do two graphs meet? At those points both equations are true at once, so the $y$-values are equal. Set the two expressions equal, solve for $x$, then find each $y$.
Find analytically the points of intersection between the parabola $y = x^2 - 4x + 3$ and the line $y = -x + 3$.
Find analytically the points of intersection between the parabola $y = x^2 - 4x + 3$ and the line $y = -x + 3$.
Given the parabolas $y = x^2 - 2x - 3$ and $y = -x^2 + 4x + 5$, find analytically the exact points where they intersect. Then calculate the straight-line distance…
Given the parabolas $y = x^2 - 2x - 3$ and $y = -x^2 + 4x + 5$, find analytically the exact points where they intersect. Then calculate the straight-line distance between the vertices of both parabolas.
Absolute value and piecewise functions
The absolute value $|f(x)|$ keeps the parts of the graph that are on or above the x-axis and reflects the parts below it upward. Zeros stay put. The result is no longer a parabola: it has corners at the x-intercepts.
A piecewise function uses different rules on different intervals. It is continuous at a breakpoint if the pieces meet there, with no jump: both rules give the same value at the breakpoint.
Draw a sketch of the graph of the function $f(x) = |x^2 - 6x + 5|$. To do this, first find the vertex and the points of intersection with the axes of the inner parabola…
Draw a sketch of the graph of the function $f(x) = |x^2 - 6x + 5|$. To do this, first find the vertex and the points of intersection with the axes of the inner parabola without the absolute value, and explain analytically how the absolute value affects the final representation.
Given the following piecewise function: $f(x) = \begin{cases} -x^2 + 4x & \text{if } x \le 3 \\ mx + n & \text{if } x > 3 \end{cases}$ Determine the values of the…
Given the following piecewise function:
$f(x) = \begin{cases} -x^2 + 4x & \text{if } x \le 3 \\ mx + n & \text{if } x > 3 \end{cases}$
Determine the values of the parameters $m$ and $n$ so that the function is continuous at the breakpoint $x = 3$ and, in addition, the half-line passes through the point $(5, -1)$.