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This is a€?Graph Using the y-Intercept and Slopea€?, section 3.4 from the book Beginning Algebra (v.

This content was accessible as of December 29, 2012, and it was downloaded then by Andy Schmitz in an effort to preserve the availability of this book. PDF copies of this book were generated using Prince, a great tool for making PDFs out of HTML and CSS. For more information on the source of this book, or why it is available for free, please see the project's home page. DonorsChoose.org helps people like you help teachers fund their classroom projects, from art supplies to books to calculators.

The steepness of any incline can be measured as the ratio of the vertical change to the horizontal change. In mathematics, we call the incline of a line the slopeThe incline of a line measured as the ratio of the vertical change to the horizontal change, often referred to as rise over run. Here we have a negative slope, which means that for every 4 units of movement to the right, the vertical change is 3 units downward.

Reading the graph from left to right, we see that lines with an upward incline have positive slopes and lines with a downward incline have negative slopes. Calculating the slope can be difficult if the graph does not have points with integer coordinates.

This leads us to the slope formulaGiven two points (x1,a€‰y1) and (x2,a€‰y2), then the slope of the line is given by the formula m=riserun=y2a?’y1x2a?’x1.. Solution: Given (a?’3,A a?’5) and (2,A 1), calculate the difference of the y-values divided by the difference of the x-values. Certainly the graph is optional; the beauty of the slope formula is that we can obtain the slope, given two points, using only algebra.

When considering the slope as a rate of change it is important to include the correct units. Example 6: A Corvette Coupe was purchased new in 1970 for about $5,200 and depreciated in value over time until it was sold in 1985 for $1,300.

Solution: Notice that the value depends on the age of the car and that the slope measures the rate in dollars per year. To this point, we have learned how to graph lines by plotting points and by using the x- and y-intercepts. Solution: In this form, the coefficient of x is the slope, and the constant is the y-value of the y-intercept. Example 8: Express 3x+5y=30 in slope-intercept form and then identify the slope and y-intercept.

Once the equation is in slope-intercept form, we immediately have one point to plot, the y-intercept.

Starting from the point (0,A 6), use the slope to mark another point 3 units down and 5 units to the right. It is not necessary to check that the second point, (5,A 3), solves the original linear equation. Marking off the slope in this fashion produces as many ordered pair solutions as we desire. Solution: In this example, we outline the general steps for graphing a line using slope-intercept form.

In this example, we notice that we could get the x-intercept by marking off the slope in a different but equivalent manner.

We could obtain another point on the line by marking off the equivalent slope down 1 unit and left 2 units.

Marking off the slope multiple times is not necessarily always going to give us the x-intercept, but when it does, we obtain a valuable point with little effort. Given any two points on a line, we can algebraically calculate the slope using the slope formula, m=riserun=y2a?’y1x2a?’x1. Any nonvertical line can be written in slope-intercept form, y=mx+b, from which we can determine, by inspection, the slope m and y-intercept (0,A b). We can obtain as many points on the line as we wish by marking off the slope multiple times. 27.A commercial van was purchased new for $20,000 and is expected to be worth $4,000 in 8 years. 28.A commercial-grade copy machine was purchased new for $4,800 and will be considered worthless in 6 years.

Express the given linear equation in slope-intercept form and identify the slope and y-intercept.

Sarah has taught secondary math and English in three states, and is currently living and working in Ontario, Canada.

Add important lessons to your Custom Course, track your progress, and achieve your study goals faster. In this lesson, we will define a parabola, discover its everyday uses, determine the functions of the intercept form of the equation and find the x-intercepts of a parabola.

DefinitionsParabola A parabola is a U-shaped graph that always has an x-squared term in its equation.

Intercept Form of a ParabolaThere are many forms of the parabola, such as vertex form, standard form and intercept form.

Lesson SummaryThe intercept form of a parabola is y = a (x - r)(x - s), where r and s are the x-intercepts, or where the graph passes through the x-axis. Please use this form if you would like to have this math solver on your website, free of charge. It's good news for any school, teacher or student when such a fantastic software is specifically designed for Algebra.

There are so many similar programs available on the market, but I was looking for something which can interact with me like a human tutor does.

I really liked the ability to choose a particular transformation to perform, rather than blindly copying the solution process.. See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and do make it available to everyone else under the same terms. However, the publisher has asked for the customary Creative Commons attribution to the original publisher, authors, title, and book URI to be removed. A positive rise corresponds to a vertical change up and a negative rise corresponds to a vertical change down.

Since subtraction is not commutative, take care to be consistent when subtracting the coordinates.

You must subtract the coordinates of the first point from the coordinates of the second point for both the numerator and the denominator in the same order. At this point, the car was beginning to be considered a classic and started to increase in value. When it is given in standard form, you have to first solve for y to obtain slope-intercept form. To do this, apply the properties of equality to first isolate 5y and then divide both sides by 5. Notice that if we mark off the slope again, from the point (5,A 3), then we obtain the x-intercept, (10,A 0).

In fact, it is a good practice to mark off the slope multiple times; doing so allows you to obtain more points on the line and produce a more accurate graph.

A positive rise denotes a vertical change up, and a negative rise denotes a vertical change down.

First, plot the y-intercept, and from this point use the slope as rise over run to mark another point on the line. The graph will open either up (smiley face) or down (sad face), and the vertex will be the lowest point if it opens up and the highest point if it opens down. The benefit of using the intercept form is that you can easily find the x-intercepts without factoring or using the quadratic formula. You may also download a PDF copy of this book (81 MB) or just this chapter (23 MB), suitable for printing or most e-readers, or a .zip file containing this book's HTML files (for use in a web browser offline).

The vertical change is called the riseThe vertical change between any two points on a line. A positive run denotes a horizontal change to the right and a negative run corresponds to a horizontal change to the left. Given any two points (x1,a€‰y1) and (x2,a€‰y2), we can obtain the rise and run by subtracting the corresponding coordinates.

In this section, we outline a process to easily determine two points using the y-intercept and the slope. A positive run denotes a horizontal change right, and a negative run denotes a horizontal change left. Finally, draw a line through these two points with a straightedge and add an arrow on either end to indicate that it extends indefinitely.

It can also demonstrate the shape of a bridge, roller coaster, arch or even the path of water in a drinking fountain. For example, vertex form helps us determine the vertex by looking at the equation, standard form helps us see the y-intercept without graphing, and intercept form helps us find the x-intercepts of the graph without factoring or using the quadratic formula. This happens in the intercept form because the equation is x - r and x - s, where r and s are the x - intercepts. Remember that the intercepts will always be the opposite sign since the formula is x - r and x - s.

Given the graph, we can calculate the slope by determining the vertical and horizontal changes between any two points. The equation of any nonvertical line can be written in slope-intercept formAny nonvertical line can be written in the form y=mx+b, where m is the slope and (0,A b) is the y-intercept. When the vertex is the lowest point, it is called a minimum, and when it is the highest point, it is called a maximum. The intercept form is y = a(x - r)(x - s), where r and s are the x-intercepts on the graph. Since we are subtracting r and s, it would make sense that that the actual intercepts are the opposite sign. Line of Symmetry The line of symmetry is a vertical line that passes through the vertex and cuts the parabola in half.

The intercept form will tell us if there are two x-intercepts, one x-intercept or no x-intercepts. Any image on one side of the parabola has a mirror image on the other side of the line of symmetry.

This content was accessible as of December 29, 2012, and it was downloaded then by Andy Schmitz in an effort to preserve the availability of this book. PDF copies of this book were generated using Prince, a great tool for making PDFs out of HTML and CSS. For more information on the source of this book, or why it is available for free, please see the project's home page. DonorsChoose.org helps people like you help teachers fund their classroom projects, from art supplies to books to calculators.

The steepness of any incline can be measured as the ratio of the vertical change to the horizontal change. In mathematics, we call the incline of a line the slopeThe incline of a line measured as the ratio of the vertical change to the horizontal change, often referred to as rise over run. Here we have a negative slope, which means that for every 4 units of movement to the right, the vertical change is 3 units downward.

Reading the graph from left to right, we see that lines with an upward incline have positive slopes and lines with a downward incline have negative slopes. Calculating the slope can be difficult if the graph does not have points with integer coordinates.

This leads us to the slope formulaGiven two points (x1,a€‰y1) and (x2,a€‰y2), then the slope of the line is given by the formula m=riserun=y2a?’y1x2a?’x1.. Solution: Given (a?’3,A a?’5) and (2,A 1), calculate the difference of the y-values divided by the difference of the x-values. Certainly the graph is optional; the beauty of the slope formula is that we can obtain the slope, given two points, using only algebra.

When considering the slope as a rate of change it is important to include the correct units. Example 6: A Corvette Coupe was purchased new in 1970 for about $5,200 and depreciated in value over time until it was sold in 1985 for $1,300.

Solution: Notice that the value depends on the age of the car and that the slope measures the rate in dollars per year. To this point, we have learned how to graph lines by plotting points and by using the x- and y-intercepts. Solution: In this form, the coefficient of x is the slope, and the constant is the y-value of the y-intercept. Example 8: Express 3x+5y=30 in slope-intercept form and then identify the slope and y-intercept.

Once the equation is in slope-intercept form, we immediately have one point to plot, the y-intercept.

Starting from the point (0,A 6), use the slope to mark another point 3 units down and 5 units to the right. It is not necessary to check that the second point, (5,A 3), solves the original linear equation. Marking off the slope in this fashion produces as many ordered pair solutions as we desire. Solution: In this example, we outline the general steps for graphing a line using slope-intercept form.

In this example, we notice that we could get the x-intercept by marking off the slope in a different but equivalent manner.

We could obtain another point on the line by marking off the equivalent slope down 1 unit and left 2 units.

Marking off the slope multiple times is not necessarily always going to give us the x-intercept, but when it does, we obtain a valuable point with little effort. Given any two points on a line, we can algebraically calculate the slope using the slope formula, m=riserun=y2a?’y1x2a?’x1. Any nonvertical line can be written in slope-intercept form, y=mx+b, from which we can determine, by inspection, the slope m and y-intercept (0,A b). We can obtain as many points on the line as we wish by marking off the slope multiple times. 27.A commercial van was purchased new for $20,000 and is expected to be worth $4,000 in 8 years. 28.A commercial-grade copy machine was purchased new for $4,800 and will be considered worthless in 6 years.

Express the given linear equation in slope-intercept form and identify the slope and y-intercept.

Sarah has taught secondary math and English in three states, and is currently living and working in Ontario, Canada.

Add important lessons to your Custom Course, track your progress, and achieve your study goals faster. In this lesson, we will define a parabola, discover its everyday uses, determine the functions of the intercept form of the equation and find the x-intercepts of a parabola.

DefinitionsParabola A parabola is a U-shaped graph that always has an x-squared term in its equation.

Intercept Form of a ParabolaThere are many forms of the parabola, such as vertex form, standard form and intercept form.

Lesson SummaryThe intercept form of a parabola is y = a (x - r)(x - s), where r and s are the x-intercepts, or where the graph passes through the x-axis. Please use this form if you would like to have this math solver on your website, free of charge. It's good news for any school, teacher or student when such a fantastic software is specifically designed for Algebra.

There are so many similar programs available on the market, but I was looking for something which can interact with me like a human tutor does.

I really liked the ability to choose a particular transformation to perform, rather than blindly copying the solution process.. See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and do make it available to everyone else under the same terms. However, the publisher has asked for the customary Creative Commons attribution to the original publisher, authors, title, and book URI to be removed. A positive rise corresponds to a vertical change up and a negative rise corresponds to a vertical change down.

Since subtraction is not commutative, take care to be consistent when subtracting the coordinates.

You must subtract the coordinates of the first point from the coordinates of the second point for both the numerator and the denominator in the same order. At this point, the car was beginning to be considered a classic and started to increase in value. When it is given in standard form, you have to first solve for y to obtain slope-intercept form. To do this, apply the properties of equality to first isolate 5y and then divide both sides by 5. Notice that if we mark off the slope again, from the point (5,A 3), then we obtain the x-intercept, (10,A 0).

In fact, it is a good practice to mark off the slope multiple times; doing so allows you to obtain more points on the line and produce a more accurate graph.

A positive rise denotes a vertical change up, and a negative rise denotes a vertical change down.

First, plot the y-intercept, and from this point use the slope as rise over run to mark another point on the line. The graph will open either up (smiley face) or down (sad face), and the vertex will be the lowest point if it opens up and the highest point if it opens down. The benefit of using the intercept form is that you can easily find the x-intercepts without factoring or using the quadratic formula. You may also download a PDF copy of this book (81 MB) or just this chapter (23 MB), suitable for printing or most e-readers, or a .zip file containing this book's HTML files (for use in a web browser offline).

The vertical change is called the riseThe vertical change between any two points on a line. A positive run denotes a horizontal change to the right and a negative run corresponds to a horizontal change to the left. Given any two points (x1,a€‰y1) and (x2,a€‰y2), we can obtain the rise and run by subtracting the corresponding coordinates.

In this section, we outline a process to easily determine two points using the y-intercept and the slope. A positive run denotes a horizontal change right, and a negative run denotes a horizontal change left. Finally, draw a line through these two points with a straightedge and add an arrow on either end to indicate that it extends indefinitely.

It can also demonstrate the shape of a bridge, roller coaster, arch or even the path of water in a drinking fountain. For example, vertex form helps us determine the vertex by looking at the equation, standard form helps us see the y-intercept without graphing, and intercept form helps us find the x-intercepts of the graph without factoring or using the quadratic formula. This happens in the intercept form because the equation is x - r and x - s, where r and s are the x - intercepts. Remember that the intercepts will always be the opposite sign since the formula is x - r and x - s.

Given the graph, we can calculate the slope by determining the vertical and horizontal changes between any two points. The equation of any nonvertical line can be written in slope-intercept formAny nonvertical line can be written in the form y=mx+b, where m is the slope and (0,A b) is the y-intercept. When the vertex is the lowest point, it is called a minimum, and when it is the highest point, it is called a maximum. The intercept form is y = a(x - r)(x - s), where r and s are the x-intercepts on the graph. Since we are subtracting r and s, it would make sense that that the actual intercepts are the opposite sign. Line of Symmetry The line of symmetry is a vertical line that passes through the vertex and cuts the parabola in half.

The intercept form will tell us if there are two x-intercepts, one x-intercept or no x-intercepts. Any image on one side of the parabola has a mirror image on the other side of the line of symmetry.

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