Graph of $y=\text{constant}*x$

Question

Graph of $y=x$ , $y=\frac{1}{2}x$ and $y=\frac{1}{4}x$ is

All are straight line. But with different slope. Why?

Answer 1

Recall the definition of the slope of a line. The slope of a line that passes through the points $(x_1, y_1)$ and $(x_2, y_2)$ is

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Since we cannot divide by zero, the slope is only defined if $x_1 \neq x_2$, that is, the slope is only defined if a line is not vertical.

Suppose that a line non-vertical passes through the point $(x_0, y_0)$. Let $(x, y)$ be any other point on the line, then $x \neq x_0$. Therefore, the slope of the line is

$$m = \frac{y - y_0}{x - x_0}$$

Multiplying both sides of the equation by $x - x_0$ yields

$$m(x - x_0) = y - y_0$$

Since equality is symmetric,

$$y - y_0 = m(x - x_0)$$

The equation $y - y_0 = m(x - x_0)$ is known as the point-slope form of the equation of a line since the line passes through the point $(x_0, y_0)$ and has slope $m$.

Each of the lines $y = x$, $y = \frac{1}{2}x$ and $y = \frac{1}{4}x$ passes through the origin, as you can verify by substituting $0$ for $x$. Hence, we can set $(x_0, y_0) = (0, 0)$, which yields

$$y = mx$$

Notice that the coefficient of $x$ is the slope. Thus, by varying the coefficient of $x$, you vary the slope.

• The line $y = x = 1x$ has slope $1$, so it rises $1$ unit for each unit you move to the right.
• The line $y = \dfrac{1}{2}x$ has slope $1/2$, so it rises $1$ unit for every two units you move to the right.
• The line $y = \dfrac{1}{4}x$ has slope $1/4$, so it rises $1$ unit for every four units you move to the right.

Answer 2

Pick a point $t$ on horizontal axis, and look at it's value $f(t)$ . As you add a non zero number to $t$ to the horizontal axis, say $1$, if you look at the value in that point $f(t+1)$, you are only adding a multiple of that number to vertical axis - $f(t+1)=c*(t+1)=c*t + c*1$. If the constant is $0$, you are not adding anything as you go along the horizontal axis, and the slope will just be horizontal. If the constant is $1$, then you are adding the same amount to the vertical axis as you added to the horizontal axis, so the slope will be diagonal.

The graphs are straight lines, because what I wrote above is true in every point- the slope is the same in every point, and is determined by the constant.