The Linear Function y = mx + b
A taxi charges 35 pesos when you get in (the flag drop) and 8 pesos per kilometer. A gym charges 300 pesos to sign up and 400 per month. Your piggy bank grows 50 pesos every week. In all of these cases the graph is a line, and with two numbers (how much it goes up per unit and where it starts) you know everything.
Table, graph, and algebraic expression
Start with a situation: your piggy bank already has 200 pesos and every week you add 50 from your allowance. Call $x$ the number of weeks that pass and $y$ the pesos you have. The same relationship can be seen in three ways, and in all three the same thing shows up: every week, 50 more pesos.
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Table: when $x$ increases by 1, $y$ always increases by the same amount (+50). The rate of change is constant.
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Graph: the points are aligned: each step of 1 to the right goes up 50. It's a line.
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Algebraic expression: $y = 50x + 200$. The 50 multiplies $x$ (what changes per week) and the 200 is what you have at the start ($x = 0$).
A fruit stand at the local market sells oranges for $\$36$ per kilogram. a) Complete a table in your notebook with the price of $1$, $2$, $3$, $4$ and $5…
A fruit stand at the local market sells oranges for $\$36$ per kilogram.
- a) Complete a table in your notebook with the price of $1$, $2$, $3$, $4$ and $5$ kg of oranges.
- b) Write the equation of the function that relates the weight $x$ in kilograms to the price $y$ in pesos.
- c) Is it a direct proportion function? What is its slope and what does it mean?
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. a) Complete the table of values in…
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.
- a) Complete the table of values in your notebook for $x = 0$, $0.5$, $1$, $1.5$ and $2$.
- b) Write the equation of the function.
- 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?
The table shows the cost of printing photographs at a print shop depending on the number of copies: | No. of copies $x$ | $10$ | $20$ | $30$ | $40$ |…
The table shows the cost of printing photographs at a print shop depending on the number of copies:
| No. of copies $x$ | $10$ | $20$ | $30$ | $40$ |
|---|---|---|---|---|
| Cost $y$ ( pesos) | $120$ | $200$ | $280$ | $360$ |
- a) Check that the cost always increases by the same amount for every $10$ copies.
- b) What is the gradient of the function? And the y-intercept?
- c) Write the equation relating $x$ and $y$.
- d) What does the y-intercept represent in this context?
Slope and y-intercept
Move $m$ and $b$. Notice that $m$ changes the steepness and $b$ raises or lowers the line without turning it:
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$m > 0$: increasing. $\quad m < 0$: decreasing. $\quad m = 0$: horizontal line (constant function).
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The larger $|m|$ is, the steeper the line.
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: a) $y = 4x - 3$ b)…
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:
- a) $y = 4x - 3$
- b) $y = -2x + 7$
- c) $y = \frac{1}{3}x$
- d) $y = 5 - x$
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 (…
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.
- a) $y = -3x$
- b) $y = x^2 + 1$
- c) $y = \frac{x}{2} - 4$
- d) $y = \frac{2}{x}$
- e) $y = 7$
The figure shows three lines, $a$, $b$ and $c$, as shown below. a) Match each line with its equation from these three: $y = 2x + 1$, $y = -x + 3$,…
The figure shows three lines, $a$, $b$ and $c$, as shown below.
- a) Match each line with its equation from these three: $y = 2x + 1$, $y = -x + 3$, $y = \frac{1}{2}x - 2$.
- b) Explain how you used the sign of the gradient and the y-intercept to decide.
Calculating the slope
If you know two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the line:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{lo que sube}}{\text{lo que avanza}}$$Calculate the gradient of the line passing through each pair of points: a) $A(2, 5)$ and $B(6, 13)$ b) $P(-1, 4)$ and $Q(3, -8)$ c) $M(0, -2)$ and $N(5, -2)$
Calculate the gradient of the line passing through each pair of points:
- a) $A(2, 5)$ and $B(6, 13)$
- b) $P(-1, 4)$ and $Q(3, -8)$
- 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: a) Calculate the gradient $m$. b) Substitute the coordinates…
Find the equation of the line passing through the points $A(1, 2)$ and $B(3, 8)$ by following these steps:
- a) Calculate the gradient $m$.
- b) Substitute the coordinates of one of the points into $y = mx + n$ to find $n$.
- 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. a) Find two points with integer coordinates that the line passes through. b) Calculate its gradient…
The line in the figure represents a linear function, as shown below.
- a) Find two points with integer coordinates that the line passes through.
- b) Calculate its gradient using those points.
- c) Write the equation of the function.
Graphing a line
Two points are enough. Make a table with two easy values of $x$ (for example, 0 and 1, or 0 and another one that gives whole numbers), calculate their $y$ values, plot the points, and connect them.
To check whether a point is on the line, substitute: $(1, 4)$ is on $y = -2x + 6$ because $-2 + 6 = 4$ ✓.
Check, by substituting the coordinates into the equation, whether the points $A(3, 4)$ and $B(-2, 11)$ lie on the line $y = -3x + 5$.
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…
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 figure shows the graph of a linear function whose y-intercept is not zero. a) What is its y-intercept? b) By how much does $y$ increase when $x$ increases by one…
The figure shows the graph of a linear function whose y-intercept is not zero.
- a) What is its y-intercept?
- b) By how much does $y$ increase when $x$ increases by one unit? What is the slope?
- c) Write its equation.
Lines in real life
In problems, $b$ is usually the fixed value (the sign-up fee, the flag drop) and $m$ is the value per unit (pesos per month, pesos per kilometer).
A gym charges a registration fee of $\$400$ and $\$300$ per month. a) Write the equation giving the total cost $y$ in pesos as a function of the number of months $x$. b)…
A gym charges a registration fee of $\$400$ and $\$300$ per month.
- a) Write the equation giving the total cost $y$ in pesos as a function of the number of months $x$.
- b) What is the y-intercept and what does it represent in this context?
- c) What about the slope? What does it mean?
A taxi in Guadalajara charges a base fare of $\$70$ plus $\$24$ per kilometer traveled. a) Write the function giving the price $y$ for a journey of $x$ kilometers. b)…
A taxi in Guadalajara charges a base fare of $\$70$ plus $\$24$ per kilometer traveled.
- a) Write the function giving the price $y$ for a journey of $x$ kilometers.
- b) How much will a journey of $12\ \text{km}$ cost?
- c) If a journey cost $\$334$, how many kilometers were traveled?
Find the point of intersection of the lines $y = 2x + 1$ and $y = -x + 7$: a) Set up the equation that results from equating the two expressions and solve it. b)…
Find the point of intersection of the lines $y = 2x + 1$ and $y = -x + 7$:
- a) Set up the equation that results from equating the two expressions and solve it.
- b) Calculate the y-coordinate of the intersection point.
- c) Check the result by substituting into both equations.
To convert degrees Celsius to degrees Fahrenheit, the function $F = 1.8C + 32$ is used. a) How many degrees Fahrenheit is $25\ \text{°C}$? b) What does the coefficient…
To convert degrees Celsius to degrees Fahrenheit, the function $F = 1.8C + 32$ is used.
- a) How many degrees Fahrenheit is $25\ \text{°C}$?
- b) What does the coefficient $1.8$ mean? And the constant term $32$?
- c) The forecast announces $86\ \text{°F}$ in a city in the United States. How many degrees Celsius is that?