The linear function $y=mx+b$ actually represents a graph of a straight line that has a point of intersection with the vertical $Y$ axis.

$m$ represents the slope. When $m$ is positive, the slope is positive: the line goes upwards. When $m$ is negative, the slope is negative: the line goes downwards. When $m = 0$, the slope is zero: the line is parallel to the $X$ axis.

$b$ represents the point where the line intersects the $Y$ axis. If $b=0$, then the line will pass through the origin of the coordinates, that is, the point $\left(0,0\right)$

If we are given a point, we can place it into the equation of a line to see if the equation holds true. If we are given just one part of the point: $X$ or $Y$, we will put the given value into the equation correctly and find the second part of the point.

How do we graph the function?

If we want a precise drawing, we'll build a table of values with $3$ or fewer values. We replace $X$ each time and obtain the value of $Y$. We consider the slope of the function to be increasing, decreasing, or equal to $0$, and then we graph it.

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To calculate the slope, we can use a formula that finds it using two given points that the line passes through:

$m=\frac{\left(Y2-Y1\right)}{(X2-X1)}$

A Lesson on Linear Functions

We are given a linear function $y=3x+4$.

We are asked to interpret the values $3$ and $4$ and plot the graph of the function.

First, it appears that $m=3$, meaning $3$ represents the slope of the line (or function).

$b=4$ means that the line intersects the vertical axis (the y-axis) at $4$.

To plot the graph, all we need are $2$ points. We substitute values and obtain:

Now we will mark the two points on the coordinate system and connect them. Looking at the graph, we can confirm that the plot intersects the y-axis at the value of $4$.

Examples and exercises with solutions for the linear function

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Find the slope of the line that passes through the points $(4,1),(2,5)$

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Remember the formula to calculate the slope using the points:

Now, replace the data in the formula:

$\frac{(5-1)}{(2-4)}=\frac{4}{-2}=-2$

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Do you know what the answer is?

Question 1

The line passes through the points \( (-2,3),(0,1) \)