Exercises: Linear and affine functions

For each of the following functions, state the value of the gradient and the y-intercept, and indicate whether its graph is increasing or decreasing:

1. a) $y = 4x - 3$

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

1. c) $y = \frac{1}{3}x$

1. d) $y = 5 - x$

A fruit stand at the local market sells oranges for $\$36$ per kilogram.

1. a) Complete a table in your notebook with the price of $1$, $2$, $3$, $4$ and $5$ kg of oranges.

1. b) Write the equation of the function that relates the weight $x$ in kilograms to the price $y$ in pesos.

1. c) Is it a direct proportion function? What is its slope and what does it mean?

The line in the figure represents a linear function, as shown below.

1. a) Find two points with integer coordinates that the line passes through.

1. b) Calculate its gradient using those points.

1. c) Write the equation of the function.

Calculate the gradient of the line passing through each pair of points:

1. a) $A(2, 5)$ and $B(6, 13)$

1. b) $P(-1, 4)$ and $Q(3, -8)$

1. c) $M(0, -2)$ and $N(5, -2)$

Of the following equations, state which correspond to a direct proportion function ($y = mx$), which to a linear function with a y-intercept other than zero ($y = mx + b$, $b \neq 0$), and which are not linear functions at all. Justify your answer.

1. a) $y = -3x$

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

1. c) $y = \frac{x}{2} - 4$

1. d) $y = \frac{2}{x}$

1. e) $y = 7$

The figure shows the graph of a linear function whose y-intercept is not zero.

1. a) What is its y-intercept?

1. b) By how much does $y$ increase when $x$ increases by one unit? What is the slope?

1. c) Write its equation.

A gym charges a registration fee of $\$400$ and $\$300$ per month.

1. a) Write the equation giving the total cost $y$ in pesos as a function of the number of months $x$.

1. b) What is the y-intercept and what does it represent in this context?

1. c) What about the slope? What does it mean?

The figure shows three lines, $a$, $b$ and $c$, as shown below.

1. a) Match each line with its equation from these three: $y = 2x + 1$, $y = -x + 3$, $y = \frac{1}{2}x - 2$.

1. b) Explain how you used the sign of the gradient and the y-intercept to decide.

Check, by substituting the coordinates into the equation, whether the points $A(3, 4)$ and $B(-2, 11)$ lie on the line $y = -3x + 5$.

A cyclist travels at a constant speed of $24\ \text{km/h}$. We call $x$ the time in hours and $y$ the distance covered in kilometers.

1. a) Complete the table of values in your notebook for $x = 0$, $0.5$, $1$, $1.5$ and $2$.

1. b) Write the equation of the function.

1. c) If the cyclist sets off from home, which is $6\ \text{km}$ from the starting point of the route, what would be the equation for the distance from home now? What type of function is it?

During a heatwave, a town's weather station recorded a constant temperature of $24\ \text{°C}$ throughout the early hours of the morning.

1. a) Write the equation of the function giving the temperature $y$ as a function of the time $x$ during that period.

1. b) What is its slope? What does its graph look like?

1. c) Is it a direct proportion function ($y = mx$), a linear function with a y-intercept other than zero ($y = mx + b$, $b \neq 0$), or neither? Explain your reasoning.

Copy and complete the table of values of the function $y = -2x + 6$ in your notebook for $x = -1$, $0$, $1$, $2$ and $3$, and plot the line on Cartesian axes. At what point does it cross the $X$ axis?

Find the equation of the line passing through the points $A(1, 2)$ and $B(3, 8)$ by following these steps:

1. a) Calculate the gradient $m$.

1. b) Substitute the coordinates of one of the points into $y = mx + n$ to find $n$.

1. c) Write the final equation and check that the other point also satisfies it.

The graph shows the balance of Adrián's digital savings account over the last few weeks, as shown in the figure.

Which of these three formulas corresponds to the graph? Justify your choice by ruling out the other two.

• $y = 100x + 200$

• $y = -100x + 800$

• $y = 200x - 100$

A taxi in Guadalajara charges a base fare of $\$70$ plus $\$24$ per kilometer traveled.

1. a) Write the function giving the price $y$ for a journey of $x$ kilometers.

1. b) How much will a journey of $12\ \text{km}$ cost?

1. c) If a journey cost $\$334$, how many kilometers were traveled?

Two hikers leave at the same time from two mountain shelters separated by $18\ \text{km}$ and walk towards each other along the same path. The first walks at $5\ \text{km/h}$ and the second at $4\ \text{km/h}$, as shown in the figure.

1. a) Write the equation giving each hiker's distance from the first shelter as a function of time $x$ in hours.

1. b) Draw both lines in your notebook.

1. c) After how much time do they cross? At what distance from the first shelter?

The graph shows the price of a parking lot as a function of time, as shown in the figure.

1. a) How much does it cost to park for $4$ hours?

1. b) What is the fixed entry fee and how much does each hour cost?

1. c) Write the equation of the function for $x \ge 0$.

Find the point of intersection of the lines $y = 2x + 1$ and $y = -x + 7$:

1. a) Set up the equation that results from equating the two expressions and solve it.

1. b) Calculate the y-coordinate of the intersection point.

1. c) Check the result by substituting into both equations.

To convert degrees Celsius to degrees Fahrenheit, the function $F = 1.8C + 32$ is used.

1. a) How many degrees Fahrenheit is $25\ \text{°C}$?

1. b) What does the coefficient $1.8$ mean? And the constant term $32$?

1. c) The forecast announces $86\ \text{°F}$ in a city in the United States. How many degrees Celsius is that?

The water tank of an irrigation community contains $1,200\ \text{m}^3$ and is consumed at a rate of $75\ \text{m}^3$ per day, as shown in the figure.

1. a) Write the function giving the water volume $y$ as a function of day $x$.

1. b) What is the gradient? Why is it negative?

1. c) What volume will remain after $9$ days?

1. d) On which day will the tank run out if it does not rain?

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

1. a) "Every direct proportion function ($y = mx$) is also a linear function of the form $y = mx + b$."

1. b) "If a line has a negative slope, then it passes through the origin."

1. c) "The lines $y = 3x + 2$ and $y = 3x - 5$ are parallel."

1. d) "The function $y = 4$ has no slope."

The table shows the cost of printing photographs at a print shop depending on the number of copies:

1. a) Check that the cost always increases by the same amount for every $10$ copies.

1. b) What is the gradient of the function? And the y-intercept?

1. c) Write the equation relating $x$ and $y$.

1. d) What does the y-intercept represent in this context?

In the school charity race, Nora gives her younger brother a head start: he starts $40\ \text{m}$ ahead and runs at $3\ \text{m/s}$, while Nora runs at $4.5\ \text{m/s}$, as shown in the figure.

1. a) Write the function giving the distance from the start in terms of time for each runner.

1. b) After how many seconds does Nora catch up with her brother?

1. c) How far from the start does this happen?

1. d) If the race ends at $150\ \text{m}$, does Nora win or does her brother reach the finish line first? Justify your answer with calculations.

Una compañía de telefonía ofrece dos tarifas, como se muestra en la figura (eje horizontal: minutos de llamadas al mes; eje vertical: coste mensual en pesos):

• Tarifa A: cuota fija de $\$12$ al mes y $\$0.04$ por minuto de llamadas.

• Tarifa B: sin cuota fija, a $\$0.08$ por minuto.

1. a) Write the monthly cost function in terms of the minutes used for each plan.

1. b) After how many minutes per month does plan A become cheaper?

1. c) Marta talks an average of $250$ minutes per month. Which plan is better for her, and how much does she save per year?

The function $F = 1.8C + 32$ converts degrees Celsius into degrees Fahrenheit.

1. a) There is a single temperature at which Celsius and Fahrenheit thermometers show the same numerical value. Set up the equation $C = F$ and calculate it.

1. b) Does the result make physical sense? Explain what happens at this temperature.

1. c) Plot the function in your notebook and interpret the point of intersection of the graph with the line $F = C$.

You can do it on paper and send a photo with your answer.

An online custom t-shirt shop charges $\$360$ for the logo design and $\$150$ per t-shirt. Another shop does not charge for the design, but each t-shirt costs $\$204$.

1. a) Write the function for the total cost as a function of the number of t-shirts for each shop.

1. b) The local basketball team needs $14$ t-shirts. Which shop is cheaper for them, and by how much?

1. c) For what number of t-shirts do both shops cost the same? What happens for larger orders?

Find the value of $k$ in each case:

1. a) So that the line $y = kx + 3$ passes through the point $(2, 11)$.

1. b) So that the line $y = 5x + k$ passes through the point $(-1, 2)$.

1. c) So that the line $y = kx - 4$ has the same gradient as the line passing through $(0, 1)$ and $(4, 9)$.

A student solved the exercise "find the gradient of the line passing through $A(2, 5)$ and $B(6, 13)$ and state if it is increasing" as follows:

"The gradient is $m = \frac{6 - 2}{13 - 5} = \frac{4}{8} = 0.5$. As $0.5 < 1$, the line is not very steep, so it is almost horizontal and decreasing."

1. a) Find the two errors they have made.

1. b) Calculate the correct gradient and classify the line.

1. c) Explain in your own words the relationship between the sign of the gradient and whether the function is increasing or decreasing.

Decision task. Three friends want to save for the end-of-year school trip, which costs $\$420$ per person. They already have some savings and each week add a fixed amount, as shown in the figure:

• Lucía: has $\$120$ and saves $\$15$ per week.

• Pablo: has $\$40$ and saves $\$25$ per week.

1. a) Write the function for the money saved as a function of weeks for each of them.

1. b) Who takes fewer weeks to reach $\$420$?

1. c) In which week does Pablo catch up with Lucía in saved money?

1. d) If the trip is in $16$ weeks, will both of them make it on time? How much money will each person have?

In a physics laboratory, masses are hung from a spring and its length is measured:

1. a) Check that the length increases at a constant rate for every $50\ \text{g}$ and write the function that gives the length $y$ as a function of the mass $x$.

1. b) What length will the spring have with a mass of $320\ \text{g}$?

1. c) What mass must be hung for the spring to measure $30.5\ \text{cm}$?

1. d) What is the physical meaning of the y-intercept?

1. a) Find the equation of the line passing through the points $A(-1, 7)$ and $B(2, -5)$.

1. b) Check whether the point $C(5, -17)$ is collinear with $A$ and $B$.

1. c) What about the point $D(0, 5)$? Justify your answer without plotting.

Accessibility regulations limit the gradient of ramps: an accessible ramp must not exceed a $10\ \,\%$ gradient. The gradient, expressed as a percentage, is the quotient of the vertical rise divided by the horizontal distance covered, multiplied by $100$, as shown in the figure.

1. a) A ramp rises by $0.54\ \text{m}$ over a horizontal distance of $6\ \text{m}$. Calculate its gradient as a percentage. Does it comply with regulations?

1. b) If you want to cover the same vertical height difference with a gradient of $8\ \,\%$, what minimum horizontal length must the ramp have?

1. c) Express the ramp from part a) as a function $y = mx$, taking the start of the ramp as the origin. What does $m$ mean?