The quadratic function
Throw a ball: it rises, slows down, falls. Its path is a parabola. So is the jet of water from a fountain, the cable of a suspension bridge or the graph of the area of a vegetable patch according to its width. Behind all of them is a function with $x^2$: the quadratic function.
The parabola and its features
Change $a$, $b$ and $c$ and see what each one does:
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Sign of $a$: if $a > 0$ it opens upwards (∪, minimum vertex); if $a < 0$, it opens downwards (∩, maximum vertex).
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Size of $|a|$: the larger it is, the narrower the parabola.
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$c$: 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 sketching
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$y$-axis: $(0, c)$.
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$x$-axis: solve $ax^2 + bx + c = 0$ (a quadratic equation). Two intercepts, one (the vertex touches the axis) or none, depending on the discriminant.
For example, $y = -x^2 + 2x + 3$: it opens downwards ($a = -1$), 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 $x = 3$.
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: maximum values
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 metres, 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 metres, 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 $x$ metres, 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 maximise the area. What is this maximum area?