Each line has a point where \(x = a + 1\).
\(\blacktriangleright \) For \(L_1\), the corresponding y-coordinate would be \(y = \answer {m+b}\)
\(\blacktriangleright \) For \(L_2\), the corresponding y-coordinate would be \(y = \answer {n+b}\)
slope
Linear functions exhibit a constant growth rate or constant rate of change and this is represented graphically by the slope of the function’s graph, which is a line.
Slope
Slope is a measurement of the tilt of a line.
\[ slope = \frac {rise}{run} = \frac {\Delta vertical}{\Delta horizontal} \]
If \((x_1, y_1)\) and \((x_2, y_2)\) are two points on a line, then the slope can be calculated as
\[ slope = \frac {rise}{run} = \frac {\Delta vertical}{\Delta horizontal} = \frac {y_2 - y_1}{x_2 - x_1} \]
\(m\) is a common symbol for slope in linear equations.
\(\blacktriangleright \) Slope tells us how a line is tilted.
Suppose we have a line that contains the point \((a,b)\) and slope \(m\). Let \((x,y)\) represent any other point on the line. Then,
\[ \frac {y-b}{x-a} = m \]
This can be rewritten as \(y - b = m(x - a)\). We call this the point-slope form of a line. This
equaiton can be rewritten as \(y = m(x - a) + b\), which can be viewed as a formula for a function: \(y(x) = m(x - a) + b\)
- lines are curves (collections of dots) that exhibit a constant slope.
- linear equations are satisfied by the coordinates of each dot on a line.
- linear functions are functions with a constant rate of change.
- slope is the graphical representation of rate of change.
Let \(f(x)\) be a linear function. Then, \(f(x)\) has a formula of the form \(f(x) = m(x-a) + f(a)\). \(m\) is called the rate-of-change of \(f(x)\).
Suppose we have the Cartesian plane displaying the graph of \(f(x)\) with the vertical axis, \(y\), representing \(f(x)\). Then we have \(y = f(x)\), which gives us \(y = m(x-a) + f(a)\), which is a linear equation, which is the equation for the plotted curve - a line in this case.
- We have an algebraic object - the equation, which describes the pairs or points.
- We have a geometric object - the line, which is a collection of points.
- We have an algebraic object - the function \(f(x)\), which collects all of the pieces into a single package.
This algebraic - geometric connection connects many algebraic characteristics with graphical features.
Parallel
Parallel lines are lines with the same slope. They are graphs of linear functions with the same rate of change.
Let \(f(x) = \frac {1}{2} x - 4\) and \(g(x) = \frac {1}{2} x + 1\).
Below are the graphs of \(y = f(x)\) and \(y = g(x)\).
![[Picture]](constantGrowthRate-9da77bb8dc278fd512d222b99cca0ec6.svg)
If two linear functions have the same rate of change, then they must differ by only a constant.
In the example above, \(f(x) - g(x) = -5\) or \(f(x) = g(x) - 5\).
Graphically, this means that either line can be shifted vertically by \(5\) units to land on the other.
Perpendicular
Prependicular lines are lines that form a right angle.
Below are the graphs of two lines, \(L_1\) and \(L_2\), whch intersect at the point \((a,b)\).
![[Picture]](constantGrowthRate-96503b2a6a1af23bf31845ead137b82b.svg)
The equation of line \(L_1\) would be \(y = m(x-a) + b\) and \(L_2\) would have the equation \(y = n(x-a) + b\), where \(m\) and \(n\) are their respective slopes.
Each line above contains the point \((a, b)\). Let’s select another point on each
line.
![[Picture]](constantGrowthRate-1e3c7e4ea13e8a78da973c9b50ba38b0.svg)
Since we have a right angle, these points are corners of a right triangle. Therefore, the lengths have to satisfy the Pythagorean Theorem.
The lengths are
- lower side : \(\sqrt {(a + 1 - a)^2 + (n + b) - b)^2} = \sqrt {\answer {1 + n^2}}\)
- higher side : \(\sqrt {(a + 1 - a)^2 + (m + b) - b)^2} = \sqrt {\answer {1 + m^2}}\)
- hypotenuse : \(\answer {m-n}\)
The Pythagorean Theorem gives us
\begin{align*} \left (\sqrt {1 + n^2}\right )^2 + \left (\sqrt {1 + m^2}\right )^2 & = (m-n))^2 \\ 1 + n^2 + 1 + m^2 & = m^2 - 2 m n + n^2 \\ 2 & = -2 m n \\ 1 & = -m n \\ \frac {1}{-m} & = n \end{align*}
The slopes are negative reciprocals of each other.
- The slopes of perpendicular lines are negative reciprocals.
- The product of the slopes of perpendicular lines equals \(-1\).
Intercepts
Almost every line intersects both the horizontal and vertical axes. These points are
called the intercepts of the line. Horizontal and vertical lines have one intercept,
unless they just happen to be one of the axes.
If we think of the line as the graph of a linear function, \(f\), then the horizontal intercept looks like \((a, 0)\), which means \(f(a)=0\) and \(a\) is the zero of \(f\). In this case, our formula looks like \(f(x) = m (x-a)\) and \((x - a)\) is a factor of the formula.
The trio
- horizontal intercepts
- factors
- zeros
all represent the same idea.
The vertical intercept looks like \((0, b)\), where \(f(x) = m(x-0)+b = m \, x + b\). This is not as valuable as the horizontal
intercept, which corresponds to a zero and a factor.
Horizontal Lines
Horizontal lines have equations of the form \(y = y_0\), where the \(y\) is the name of the vertical axis. You can view this equation as \(y = 0 \cdot x + y_0\) to see that every point on this line is of the form \((x, y_0)\), making it horizontal.
Horizontal Line
![[Picture]](constantGrowthRate-a1bdaafa187533d56f0fd3f373b4a4c9.svg)
The equation of this line is \(w=3\).
Every point on this line is of the form \(\left (t_0, \answer {3}\right )\). It is the graph of the constant function \(f(t)=3\).
Horizontal lines are graphs of constant functions. Constant functions are linear
functions with a growth rate of \(0\).
Vertical Lines
Vertical lines have equations of the form \(x = x_0\), where the \(x\) is the name of the horizontal axis. You can view this equation as \(x = 0 \cdot y + x_0\) to see that every point on this line is of the form \((x_0,y)\), making it vertical.
Vertical Line
![[Picture]](constantGrowthRate-a489c91a001064cc2d9033d375a6f85f.svg)
The equation of this line is \(h=4\).
Every point on this line is of the form \(\left (\answer {4}, g_0\right )\). This graph corresponds to no linear function, because the single domain number \(4\) is paired with more than one codomain number.
Every line is the graph of a linear function, except vertical lines.
Product Form
We have many forms for writing linear equations and formulas for linear functions.
\(\blacktriangleright \) Point-Slope Form
Given a point on a line, \((a, f(a))\), and the slope, \(m\), we can write the formula as \(f(x) = m(x-a) + f(a)\). This emphasizes the behavior around \(a\).
\(\blacktriangleright \) Slope-Intercept Form
If we choose our point to be the vertical intercept \((0, b)\), then the slope-intercept form looks like \(f(x) = m \, x + f(0)\) or \(f(x) = m \, x + b\).
These are both sums.
However, our interest focuses on zeros of the function, which we would like to correspond to factors in our formulas.
As long as \(m \ne 0\), we can take our sum and transform it into a product.
\begin{align*} f(x) & = m(x-a) + f(a) \\ & = m \left (x - a + \frac {f(a)}{\answer {m}}\right ) \\ & = m \left (x - \left (a - \frac {f(a)}{m}\right )\right ) \end{align*}
Since \(f\) is linear, \(f\) has one zero. It is \(a - \frac {f(a)}{m}\). It is visually encoded as the horizontal intercept, \(\left (a - \frac {f(a)}{m}, 0\right )\).
This zero will always exist as long as \(m \ne 0\).
Factored Form
Let \(K(t) = 3t - 15\).
\(5\) is the only root of \(K\). We can rewrite this formula in factored form knowing that \(t-5\) has to be a factor.
\(K(t) = 3 \left (t - \answer {5} \right )\).
Since \(5\) is the only root of \(K(t)\), we know that \(\left (\answer {5}, \answer {0}\right )\) is the only \(t\)-intercept of the corresponding
line.
In either form, the coefficient of \(t\) is \(\answer {3}\).
This coefficent measures the rate of change of \(K\) and the slope of the line.
ooooo-=-=-=-ooOoo-=-=-=-ooooo
more examples can be found by following this link
More Examples of Linear Functions