Point Slope Form: Videos & Practice Problems

Understanding the equation of a line involves mastering both slope intercept form and point slope form. The slope intercept form, $y=mx+b$, uses the slope ($m$) and y-intercept ($b$). Point slope form, $y-y_1=m(x-x_1)$, applies when given a point and slope. Converting between these forms involves distributing the slope and isolating $y$. This skill is essential for graphing lines and solving polynomial expressions involving terms and coefficients.

Point Slope Form of a Line

Understanding how to find the equation of a line is fundamental in algebra, and two common forms used are slope-intercept form and point-slope form. The slope-intercept form is expressed as y = mx + b, where m represents the slope of the line and b is the y-intercept, the point where the line crosses the y-axis. The slope is calculated as the ratio of the rise (change in y) over the run (change in x), which determines the steepness of the line.

While slope-intercept form is straightforward when the slope and y-intercept are known, point-slope form is especially useful when you have the slope and any point on the line, not necessarily the y-intercept. The point-slope form is given by the equation:

\[y - y_1 = m(x - x_1)\]

Here, (x₁, y₁) is a specific point on the line, and m is the slope. This form allows you to write the equation of a line when you know a point it passes through and its slope.

For example, if a line has a slope of 2 and passes through the point (1, 3), the point-slope form becomes:

\[y - 3 = 2(x - 1)\]

To convert from point-slope form to slope-intercept form, distribute the slope on the right side and then solve for y. Using the previous example:

\[y - 3 = 2x - 2\]

Adding 3 to both sides gives:

\[y = 2x + 1\]

This conversion is essential for graphing and interpreting linear equations easily.

Consider another example where the slope m = \frac{1}{2} and the line passes through the point (-6, -2). Using point-slope form:

\[y - (-2) = \frac{1}{2}(x - (-6))\]

Simplifying the double negatives:

\[y + 2 = \frac{1}{2}(x + 6)\]

Distribute the slope:

\[y + 2 = \frac{1}{2}x + 3\]

Subtract 2 from both sides to solve for y:

\[y = \frac{1}{2}x + 1\]

Mastering both point-slope and slope-intercept forms enhances your ability to write and manipulate linear equations based on different given information. Recognizing when to use each form and converting between them strengthens problem-solving skills and deepens understanding of linear relationships in coordinate geometry.

Write each equation in point-slope form.
Slope: $m=4$; Point: $(1,-2)$

• A. $y-2=4(x-1)$
• B. $y-4=2(x-1)$
• C. $y+4=2(x-1)$
• D. $y+2=4(x-1)$

Answer: (D)

Write each equation in point-slope form.
Slope: $m=-\frac{3}{2}$; Point: $(0,5)$

• A. $y-5=-\frac{3}{2}x$
• B. $y=-\frac{3}{2}(x-5)$
• C. $y=\frac{3}{2}(x-5)$
• D. $y+5=-\frac{3}{2}x$

Answer: (A)

Write each equation in point-slope form.
Slope: $m=\frac{1}{3}$; Point: $(-2,4)$

• A. $y-4=\frac{1}{3}(x-2)$
• B. $y-4=\frac{1}{3}(x+2)$
• C. $y-\frac{1}{3}=4(x+2)$
• D. $y-\frac{1}{3}=4(x-2)$

Answer: (B)

Point Slope Form of a Line Example 1

Graphing a line using a given point and slope involves understanding how the slope represents the ratio of vertical change (rise) to horizontal change (run). For example, if a line passes through the point (3, -1) with a slope of 2, you first locate the point on the coordinate plane. The slope of 2 can be expressed as the fraction \(\frac{2}{1}\), meaning you rise 2 units vertically and run 1 unit horizontally. Starting at (3, -1), moving up 2 units and right 1 unit gives another point on the line. Repeating this process creates multiple points that, when connected, form the line with the correct slope passing through the given point.

In another case, if the line passes through (-2, 5) with a slope of \(-\frac{1}{4}\), the negative slope indicates a downward rise. Here, the rise is -1 (down 1 unit) and the run is 4 (right 4 units). Starting at (-2, 5), moving down 1 unit and right 4 units locates a second point on the line. Continuing this pattern helps plot the line accurately. This method of graphing lines using a point and slope is essential for visualizing linear relationships and understanding how slope affects the direction and steepness of a line.

Remember, the slope formula is given by \(m = \frac{\Delta y}{\Delta x}\), where \(m\) is the slope, \(\Delta y\) is the change in the y-values (rise), and \(\Delta x\) is the change in the x-values (run). By applying this concept, you can graph any line when provided with a point and its slope, reinforcing the connection between algebraic equations and their graphical representations.

Here’s what students ask on this topic:

What is the point slope form of a line and when should it be used?

Point slope form is an equation used to represent a line when you know a point on the line and its slope. The formula is $y-y_1=m(x-x_1)$, where $x_1$ and $y_1$ are the coordinates of the known point, and $m$ is the slope. This form is especially useful when the y-intercept is not known, but you have a point on the line and the slope. It allows you to write the equation directly without needing to find the y-intercept first. This is different from slope intercept form, which requires the slope and y-intercept. Understanding point slope form helps in graphing lines and converting between different line equations.

How do you convert an equation from point slope form to slope intercept form?

To convert from point slope form $y-y_1=m(x-x_1)$ to slope intercept form $y=mx+b$, follow these steps: first, distribute the slope $m$ across the parentheses. This gives $y-y_1=mx-m{x}_1$. Next, isolate $y$ by adding $y_1$ to both sides, resulting in $y=mx-m{x}_1+y_1$. The expression $-m{x}_1+y_1$ is the y-intercept $b$. This process helps you rewrite the equation in the more familiar slope intercept form, which is useful for graphing and understanding the line's behavior.

How do you find the slope of a line given two points?

The slope of a line measures its steepness and is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points. Given two points $(x$₁,y₁) and $(x$₂,y₂), the slope $m$ is found by the formula $m=\frac{y}{}$₂−y₁x₂−x₁. This means you subtract the y-coordinates and divide by the difference of the x-coordinates. The slope tells you how much y changes for each unit change in x. This value is essential for writing the equation of the line in point slope or slope intercept form.

What is the difference between slope intercept form and point slope form?

Slope intercept form and point slope form are two ways to write the equation of a line, but they serve different purposes. Slope intercept form is $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept, the point where the line crosses the y-axis. This form is useful when you know the slope and y-intercept. Point slope form is $y-y_1=m(x-x_1)$, used when you know the slope and a specific point on the line, but not necessarily the y-intercept. Point slope form is often a starting point to find the slope intercept form by algebraic manipulation. Both forms describe the same line but are used in different contexts depending on the information given.

How do you write the equation of a line using point slope form given a point and slope?

To write the equation of a line using point slope form, you need a point on the line, $(x$₁,y₁), and the slope $m$. The formula is $y-y_1=m(x-x_1)$. Substitute the known values of $x_1$, $y_1$, and $m$ into the equation. For example, if the point is (3, 4) and the slope is 2, the equation becomes $y-4=2(x-3)$. This equation represents the line passing through the point with the given slope.