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) $y = 5 - x$

Of the following equations, state which correspond to a linear function, which to an affine function, and which are neither linear nor affine. 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 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.

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 $N(5, -2)$

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 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.

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?

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

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?

The graph shows an affine function, as shown in the figure.

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 $\text{RD\$}\ 1\,200$ and $\text{RD\$}\ 900$ 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 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?

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.

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 $\text{RD\$}\ 12$ al mes y $\text{RD\$}\ 0.04$ por minuto de llamadas.

• Tarifa B: sin cuota fija, a $\text{RD\$}\ 0.08$ 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?