The Quadratic Function
Throw a ball: it rises, slows down, and falls. Its path is a parabola. So is the stream of water from a fountain, the cable of a suspension bridge, or the graph of the area of a garden based on its width. Behind all of them is a function with $x^2$: the quadratic function.
Table, graph, and algebraic expression
You have 20 m of fencing to enclose a rectangular pen. If the width is $x$ meters, the length is $10 - x$ (because width + length is half the perimeter). How does the area vary when you change the width? See it in all three representations.
Now the graph: the points in the table fall on the parabola $A = -x^2 + 10x$.
In free fall, the distance covered by a dropped object is approximately $d = 5t^2$, where $t$ is the time in seconds and $d$ is the distance in meters. a) Complete in…
In free fall, the distance covered by a dropped object is approximately $d = 5t^2$, where $t$ is the time in seconds and $d$ is the distance in meters.
- a) Complete in your notebook a table of values for $t = 0$, $1$, $2$, $3$ and $4$.
- b) How far has it fallen after $3\ \text{s}$?
- c) If a piece of rubble takes $2.5\ \text{s}$ to reach the ground, from what height did it fall?
The table corresponds to a quadratic function: | $x$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | |---|---|---|---|---|---|---| | $y$ | $3$ | $6$ | $13$ | $24$ | $39$ | $58$ |…
The table corresponds to a quadratic function:
| $x$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|---|
| $y$ | $3$ | $6$ | $13$ | $24$ | $39$ | $58$ |
- a) Calculate the differences between consecutive values of $y$ (first differences).
- b) Calculate the differences of those differences (second differences). What do you observe?
- c) What is the vertex of the parabola? In which direction does it open?
The parabola and its elements
Move $a$, $b$, and $c$ and observe what each one does:
-
Sign of $a$: if $a > 0$ it opens upward (∪, minimum vertex); if $a < 0$, it opens downward (∩, maximum vertex).
-
Size of $|a|$: the larger it is, the narrower the parabola.
-
$c$: it is the $y$-intercept, because $f(0) = c$.
For each of these quadratic functions, state the values of the coefficients $a$, $b$ and $c$, and reason, without plotting it, whether the parabola opens upwards or…
For each of these quadratic functions, state the values of the coefficients $a$, $b$ and $c$, and reason, without plotting it, whether the parabola opens upwards or downwards:
- a) $y = 2x^2 - 3x + 5$
- b) $y = -x^2 + 4x$
- c) $y = 6 - x^2$
- d) $y = \frac{1}{2}x^2$
The figure shows the parabolas $y = x^2$, $y = 2x^2$ and $y = \frac{1}{2}x^2$, as shown below. a) Match each parabola with its equation. b) What effect does the value of…
The figure shows the parabolas $y = x^2$, $y = 2x^2$ and $y = \frac{1}{2}x^2$, as shown below.
- a) Match each parabola with its equation.
- b) What effect does the value of the coefficient $a$ in $y = ax^2$ have on the shape of the parabola?
State, without plotting them, whether the vertex of each parabola is a maximum or a minimum of the function: a) $y = -5x^2 + 3x - 1$ b) $y = 0.2x^2 - 4x + 6$ c)…
State, without plotting them, whether the vertex of each parabola is a maximum or a minimum of the function:
- a) $y = -5x^2 + 3x - 1$
- b) $y = 0.2x^2 - 4x + 6$
- c) $y = 7x - x^2$
Explain which coefficient it depends on and why.
The vertex
Consider the parabola $y = x^2 - 4x + 3$. a) Calculate the x-coordinate of the vertex using the formula $x = \frac{-b}{2a}$ and then find its y-coordinate. b) Make a…
Consider the parabola $y = x^2 - 4x + 3$.
- a) Calculate the x-coordinate of the vertex using the formula $x = \frac{-b}{2a}$ and then find its y-coordinate.
- b) Make a table of values taking values of $x$ around the vertex.
- c) Plot it and state its axis of symmetry.
A parabola has its vertex at the point $V(1, -4)$ and passes through the point $P(3, 0)$. a) What is its axis of symmetry? b) Using the symmetry of the parabola, find…
A parabola has its vertex at the point $V(1, -4)$ and passes through the point $P(3, 0)$.
- a) What is its axis of symmetry?
- b) Using the symmetry of the parabola, find another point through which it also passes.
- c) At what points does it intersect the $X$-axis?
Intercepts with the axes and graphing
-
$y$-axis: $(0, c)$.
-
$x$-axis: solve the quadratic equation $ax^2 + bx + c = 0$, by factoring or with the quadratic formula. Two intercepts, one (the vertex touches the axis), or none, depending on the discriminant $b^2 - 4ac$.
For example, $y = -x^2 + 2x + 3$: it opens downward ($a = -1$), vertex at $x_V = \frac{-2}{-2} = 1$, $y_V = 4$; it crosses the $y$-axis at $(0, 3)$ and the $x$-axis at $x = -1$ and $x = 3$, because factoring gives $-x^2 + 2x + 3 = -(x - 3)(x + 1)$.
Consider the function $y = x^2 - 2$. a) Copy and complete in your notebook the table of values for $x = -3$, $-2$, $-1$, $0$, $1$, $2$ and $3$. b) Plot the parabola on…
Consider the function $y = x^2 - 2$.
- a) Copy and complete in your notebook the table of values for $x = -3$, $-2$, $-1$, $0$, $1$, $2$ and $3$.
- b) Plot the parabola on Cartesian axes.
- c) What is its vertex? And its axis of symmetry?
Without plotting them, calculate the point of intersection with the $Y$-axis for each of these parabolas: a) $y = x^2 - 6x + 8$ b) $y = -3x^2 + 2x - 7$ c) $y = 4x^2$…
Without plotting them, calculate the point of intersection with the $Y$-axis for each of these parabolas:
- a) $y = x^2 - 6x + 8$
- b) $y = -3x^2 + 2x - 7$
- c) $y = 4x^2$
Which coefficient of the equation always determines this point of intersection?
Find the points of intersection of the parabola $y = x^2 - 5x + 6$ with the coordinate axes: a) With the $Y$-axis. b) With the $X$-axis, by solving the equation…
Find the points of intersection of the parabola $y = x^2 - 5x + 6$ with the coordinate axes:
- a) With the $Y$-axis.
- b) With the $X$-axis, by solving the equation $x^2 - 5x + 6 = 0$.
- c) Calculate the vertex and check that its x-coordinate is exactly halfway between the two intersections with the $X$-axis.
Parabolas in real life: maximums
When a problem asks for the maximum height, the maximum area, or the maximum profit and the formula is quadratic, the answer is at the vertex.
The graph shows the height of a ball thrown upwards as a function of time, as shown in the figure. a) What is the maximum height reached by the ball and at what time? b)…
The graph shows the height of a ball thrown upwards as a function of time, as shown in the figure.
- a) What is the maximum height reached by the ball and at what time?
- b) From what height was it thrown?
- c) At what time does the ball return to its launch height?
A volleyball is hit from the ground and its height, in meters, is given by $h = -5t^2 + 20t$, where $t$ is the time in seconds, as shown in the figure. a) At what time…
A volleyball is hit from the ground and its height, in meters, is given by $h = -5t^2 + 20t$, where $t$ is the time in seconds, as shown in the figure.
- a) At what time does it reach its maximum height? What is this height?
- b) How long does it take to return to the ground?
- c) What is its height at $3\ \text{s}$? Is it going up or down?
A gardener wants to fence off a rectangular vegetable patch next to the wall of his house, so that he only needs fencing for three sides. He has $40\ \text{m}$ of…
A gardener wants to fence off a rectangular vegetable patch next to the wall of his house, so that he only needs fencing for three sides. He has $40\ \text{m}$ of fencing available.
- a) If the side perpendicular to the wall measures $x$ meters, express the length of the parallel side as a function of $x$.
- b) Write the function that gives the area of the vegetable patch as a function of $x$. What type of function is it?
- c) Calculate the vertex of the parabola and deduce the dimensions that maximize the area. What is this maximum area?