The Quadratic Function

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:

1. a) $y = 2x^2 - 3x + 5$

1. b) $y = -x^2 + 4x$

1. c) $y = 6 - x^2$

1. 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.

1. a) Match each parabola with its equation.

1. 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:

1. a) $y = -5x^2 + 3x - 1$

1. b) $y = 0.2x^2 - 4x + 6$

1. c) $y = 7x - x^2$

Explain which coefficient it depends on and why.

Consider the function $y = x^2 - 2$.

1. a) Copy and complete in your notebook the table of values for $x = -3$, $-2$, $-1$, $0$, $1$, $2$ and $3$.

1. b) Plot the parabola on Cartesian axes.

1. c) What is its vertex? And its axis of symmetry?

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.

1. a) Complete in your notebook a table of values for $t = 0$, $1$, $2$, $3$ and $4$.

1. b) How far has it fallen after $3\ \text{s}$?

1. 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:

1. a) Calculate the differences between consecutive values of $y$ (first differences).

1. b) Calculate the differences of those differences (second differences). What do you observe?

1. c) What is the vertex of the parabola? In which direction does it open?

The figure shows the graph of a parabola. Each square of the grid measures 1 unit on each axis.

1. a) What are the coordinates of the vertex? Is it a maximum or a minimum?

1. b) At which points does the parabola cross the $X$ axis?

1. c) What is its axis of symmetry?

Consider the parabola $y = x^2 - 4x + 3$.

1. a) Calculate the x-coordinate of the vertex using the formula $x = \frac{-b}{2a}$ and then find its y-coordinate.

1. b) Make a table of values taking values of $x$ around the vertex.

1. 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 $P(3, 0)$.

1. a) What is its axis of symmetry?

1. b) Using the symmetry of the parabola, find another point through which it also passes.

1. c) At what points does it intersect the $X$-axis?

A student calculates the vertex of the parabola $y = 2x^2 - 8x + 1$ as follows:

"$x = \frac{b}{2a} = \frac{-8}{2 \times 2} = -2$. For $x = -2$, $y = 2 \times (-2)^2 - 8 \times (-2) + 1 = 25$. The vertex is $V(-2, 25)$ and it is a minimum because $a = 2 > 0$."

1. a) Find the errors they have made in the formula and in the calculations.

1. b) Find the correct vertex and reason whether it is a maximum or a minimum.

1. c) Check that the value of $y$ at the vertex is consistent with the direction in which the parabola opens.

On a parabola, the points $A(-1, 5)$ and $B(5, 5)$ are at the same height.

1. a) Without knowing the equation, calculate the x-coordinate of the vertex using symmetry.

1. b) If, in addition, the parabola passes through the origin, reason what the other point of intersection with the $X$-axis is.

1. c) Find the equation of the parabola knowing that the y-coordinate of its vertex is $-4$.

Without plotting them, calculate the point of intersection with the $Y$-axis for each of these parabolas:

1. a) $y = x^2 - 6x + 8$

1. b) $y = -3x^2 + 2x - 7$

1. 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:

1. a) With the $Y$-axis.

1. b) With the $X$-axis, by solving the equation $x^2 - 5x + 6 = 0$.

1. c) Calculate the vertex and check that its x-coordinate is exactly halfway between the two intersections with the $X$-axis.

The graph in the figure corresponds to one of these three functions, as shown below:

• $y = x^2 - 4x + 5$

• $y = -x^2 + 4x - 1$

• $y = x^2 + 4x + 5$

1. a) Which of the three is it? Justify your answer using the direction of opening, the $Y$-intercept, and the axis of symmetry.

1. b) Confirm your choice by calculating the vertex of the chosen function and comparing it with the one in the graph.

State whether each statement is true or false and justify your answer with an example or a counterexample:

1. a) "Every parabola intersects the $X$-axis at two points."

1. b) "The parabola $y = x^2 + c$ never has its vertex below the point $(0, c)$."

1. c) "If two parabolas have the same vertex, then they are the same parabola."

1. d) "The graph of $y = -x^2$ is obtained by reflecting $y = x^2$ in the $X$-axis."

Consider the family of parabolas $y = x^2 - 6x + k$, where $k$ is any number.

1. a) Calculate the vertex of the parabola as a function of $k$. What do all the parabolas in the family have in common?

1. b) For what values of $k$ does the parabola intersect the $X$-axis at two points? At just one point? At no points?

1. c) Plot in your notebook the parabolas for $k = 5$, $k = 9$ and $k = 13$ and verify the above.

The figure shows three parabolas, $a$, $b$ and $c$, obtained by translating the parabola $y = x^2$.

1. a) Match each one with its equation: $y = x^2 + 3$, $y = (x - 2)^2$, $y = x^2$.

1. b) What are the coordinates of the vertex of each one?

1. c) What would be the coordinates of the vertex of $y = (x + 1)^2 - 5$? Justify your answer without graphing it.

Find the equation of the parabola $y = ax^2 + bx + c$ that passes through the points $A(0, 3)$, $B(1, 0)$ and $C(3, 0)$:

1. a) Substitute the coordinates of $A$ to obtain the value of $c$.

1. b) Substitute those of $B$ and $C$ and set up a system of two equations with unknowns $a$ and $b$.

1. c) Solve the system and write the equation of the parabola. What is its vertex?

The graph shows the height of a ball thrown upwards as a function of time, as shown in the figure.

1. a) What is the maximum height reached by the ball and at what time?

1. b) From what height was it thrown?

1. 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.

1. a) At what time does it reach its maximum height? What is this height?

1. b) How long does it take to return to the ground?

1. 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 fencing available.

1. a) If the side perpendicular to the wall measures $x$ meters, express the length of the parallel side as a function of $x$.

1. b) Write the function that gives the area of the vegetable patch as a function of $x$. What type of function is it?

1. c) Calculate the vertex of the parabola and deduce the dimensions that maximize the area. What is this maximum area?

In a basketball free throw, the ball follows a trajectory given by $h = -0.08x^2 + 0.64x + 1.9$, where $x$ is the horizontal distance to the player in meters and $h$ is the height of the ball in meters. The basket is $4.6\ \text{m}$ from the player and the hoop is at a height of $3.05\ \text{m}$.

1. a) What height is the ball at when it is directly above the basket? Does the shot go in?

1. b) What is the maximum height reached by the ball during the shot?

A residents' association wants to fence off a rectangular garden using the $120\ \text{m}$ of wire mesh available, as shown in the figure.

1. a) If one of the sides measures $x$ meters, express the other side as a function of $x$.

1. b) Write the function that gives the area of the garden as a function of $x$.

1. c) Find the dimensions that maximize the area and calculate that area.

1. d) Check that the rectangle with maximum area is a square. Do you think this will always happen when fencing with a fixed perimeter? Investigate this with a perimeter of $80\ \text{m}$.

A fireworks rocket reaches a height, in meters, given by $h = -5t^2 + 25t$ in the $t$ seconds after launch.

1. a) Set up and solve the equation that allows you to find out at what times the rocket is at a height of $30\ \text{m}$.

1. b) Explain why there are two solutions and what each one means.

1. c) What maximum height does the rocket reach?

Decision task. The school theater is selling tickets for the end-of-year play. From the experience of previous years, if a ticket costs $\text{RD\$}\ 6$, $300$ spectators attend, and for every $\text{RD\$}\ 0.50$ increase in price, $20$ fewer spectators attend, as shown in the figure.

1. a) If the price increases by $x$ times $\text{RD\$}\ 0.50$, express the ticket price and the number of spectators as a function of $x$.

1. b) Write the function that gives the revenue as a function of $x$. What type of function is it?

1. c) What ticket price maximises revenue? How much would be collected?

1. d) The parents' association proposes lowering the ticket price to $\text{RD\$}\ 5$ to fill the hall. Compare the revenue with that of the optimal case and decide which of the two options you would recommend, supporting your choice with calculations.

Investigate whether the line $y = 2x + 1$ intersects the parabola $y = x^2 - 3x + 5$:

1. a) Set up the system formed by both equations and reduce it to a single quadratic equation.

1. b) Solve it and interpret the result: do they intersect? At how many points?

1. c) Repeat the investigation with the line $y = 2x - 4$. What is the difference?

A diver jumps from a diving board and her height above the water, in meters, is given by $h = -5t^2 + 6t + 1.8$, with $t$ in seconds.

1. a) How high is the diving board? Justify where this figure comes from in the formula.

1. b) What is the maximum height reached by the diver?

1. c) Set up the equation to find out how long she takes to reach the water and solve it. Why do you discard one of the solutions?

A mountain tunnel has a parabolic cross-section: its width at the base is $12\ \text{m}$ and its maximum height at the center is $6\ \text{m}$, as shown in the figure.

1. a) Placing the origin of coordinates at the center of the road, justify that the arch of the tunnel has the equation $y = ax^2 + 6$ and find the value of $a$.

1. b) A lorry of width $4\ \text{m}$ and height $3.8\ \text{m}$ drives down the center. Does it fit through the tunnel? Calculate the height of the arch at the sides of the lorry.

1. c) If the lorry moves $1\ \text{m}$ to one side, does it still fit through? Justify your answer with calculations.