What is Point Slope Form

6 Key excerpts on "What is Point Slope Form"

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  eBook - ePub

  Making Sense of Mathematics for Teaching High School

  Understanding How to Use Functions

  • Edward C. Nolan, Juli K. Dixon, Farhsid Safi, Erhan Selcuk Haciomeroglu(Authors)
  • 2016(Publication Date)
  • Solution Tree Press

    (Publisher)

  y-intercept, the equation of a line can be easily constructed.

  How does the slope formula connect to the point-slope form of the equation of a line? Consider the slope formula . What would happen if instead of two fixed points (x1 , y1 ) and (x2 , y2 ), you examined the line created by a point (x1 , y1 ) and a generic point (x, y) anywhere on the line? If you are given the slope and one point on the line (x1 , y1 ), then the formula for the slope of the line would be . When you rearrange the new equation, you derive the point-slope form of linear functions: y − y1 = m(x − x1 ).

  What is the relationship between this form and the previous y = mx + b form? What would happen if, instead of a fixed point (x1 , y1 ), the given point were to be on the y-axis? As long as the line is not vertical, this change in point would not result in a loss of generality and you could call this point on the y-axis (0, b). What would happen if you substitute the point (0, b) in for (x1 , y1 )? The equation transitions from y − y1 = m(x − x1 ) to y − b = mx, which can be written as y = mx + b

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  eBook - ePub

  Let's Review Regents: Algebra I Revised Edition

  • Gary M. Rubinstein(Author)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  y when the other variable’s value is known.

  Graphing the Solution Set of a Linear Equation That Is in Slope-Intercept Form

  Math Facts

  An equation like y = 2x + 3 or is said to be in

  slope-intercept form

  . In general, an equation of the form y = mx + b is in slope-intercept form with the m representing the slope and the b representing the y-intercept.

  To graph the solution set of a linear equation that is already in slope–intercept form:

  1. Plot the point (0, b), which is on the y-axis. If the equation is y = 2x + 3, plot the point (0, 3). This is known as the y-intercept of the graph.
  2. If the coefficient of the x term is not already a fraction, turn it into a fraction by putting the coefficient in the numerator of a fraction and a 1 in the denominator. If the coefficient is a negative fraction, make the numerator negative and the denominator positive. For the example, y = 2x + 3, the slope, denoted by m, is 2, which gets changed into .
  3. Starting at the y-intercept you already plotted, move right the number in the denominator of the slope. Then, from where you stopped, move up (down if it is negative) the number in the numerator of the slope. For the y = 2x + 3 example, the slope is so from (0, 3), you move to the right 1 unit and then up 2 units to get to the point (1, 5).
  4. Draw a line through the y-intercept and the new point. Put arrows on both sides of the line to indicate that it continues forever on both sides.

  Example 1

  Make a sketch of the solution set of the graph using the slope-intercept process.

  Solution: Since the constant is 5, the y-intercept is (0, 5). Since the coefficient of the x term is ,

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  eBook - ePub

  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

  • Philip Brown(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  m, thus:

  We make the following observations regarding the direction in which the line leans and the value of the slope of the line:

  •If a line leans to the left, its slope is negative.

  •If a line leans to the right, its slope is positive.

  •If a line is horizontal, its slope is zero.

  •If a line is vertical, its slope is infinite.

  EXAMPLE 2.3.4. The slope of the line through the points P(−3, −2) and Q(3, 4) is

  If the slope m of a line is known, then an expression relating the coordinates of any other point (x, y) on the line to the coordinates of any given point (x1 , y1 ) on the line is Another way to write this is:

  EXAMPLE 2.3.5. To find the equation of the line through the points P(−3,−2) and Q(3, 4), we substitute m = 1 from example 2.3.4 and the coordinates of Q, that is, x1 = 3 and y1 = 4 (we could also use the coordinates of P) in formula (2.1), resulting in y − 4 = (x − 3). This can be simplified to y = x + 1. The graph of this line is shown in figure 2.6 .

  FIGURE 2.6. A line through two given points.

  If we only know the slope of a line and the value of the y-intercept of the line, then we can substitute these values into another form of the equation of a line known as:

  where c is the y-intercept of the line because this corresponds to the value of y when x = 0.

  EXAMPLE 2.3.6. If the slope of a line is m = −2 and it cuts the y-axis at c = 3, then its equation is y = −2x + 3.

  In the special case that the y-intercept is zero, the line passes through the origin. Figure 2.7 shows a number of lines passing through the origin, with the slope of each line labeled next to it.

  FIGURE 2.7. Lines through the origin.

  Another form of the equation of a line that we will take note of here is: which can be obtained by algebraic manipulation of the point-slope form or slope-intercept form and renaming the constants.

  EXAMPLE 2.3.7. The equation of the line in example 2.3.5

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  Painless Pre-Algebra

  • Amy Stahl(Author)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  y-value will always be 2. Then plot the points, and draw the line through the points. This is a horizontal line.

  x	y
  −2	2
  0	2
  3	2

  Graph x = −1.

  Make a table, but this time the x-values always stay the same. The x-value has to be −1. Choose three different y-values to help you make the graph. Plot the points, and draw the line. This is a vertical line.

  x	y
  −1	−2
  −1	0
  −1	2

  CAUTION—Major Mistake Territory!

  A horizontal line is always written in the form y = b. A vertical line is always written in the form x = a.

  Examples:

  y = 3 → Horizontal line

  x = −2 → Vertical line

  BRAIN TICKLERSSet # 17

  Graph each line.

  1.x = 4

  2.x = −2

  3.y = 5

  4.y = −1

  (Answers are on pages 117 –118 .)

  Slope of a Line

  Have you ever tried to run up a steep hill? Or ride a bike all the way up a hill without walking? Or ski down a hill? The concept of “steepness” is one that can easily be understood in the real world.

  Examples of different “slopes” are easy to see. As a beginner skier, which hill would you prefer to ski down? A beginner skier starts on the bunny hill, which is less steep than a hill considered a black diamond!

  In the world of mathematics, the word slope describes the “steepness” of a line.

  There are four basic types of slope.

  Positive slope Slants upward, from left to right	Negative slope Slants downward, from left to right
  Zero slope A horizontal line, left to right Has zero slope (like a floor)	No slope/Undefined slope A vertical line, up and down Has undefined slope (like a wall)

  The slope of a line is described as a ratio. It is the comparison of the vertical change in the line compared to the horizontal change in the line. Here is the formal definition of slope.

  For a painless way to remember slope, use this saying:

  •Rise is the vertical distance between points. Rise involves the y-

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  Precalculus

  A Self-Teaching Guide

  • Steve Slavin, Ginny Crisonino(Authors)
  • 2001(Publication Date)
  • Trade Paper Press

    (Publisher)

  m.

  When we use our slope formula it doesn’t make any difference which one we call the first point and which one we call the second point; we’ll still get the same slope when we use the slope formula . Four graphs are shown below. The line in (a) rises up, in (b) it falls, (c) is a horizontal line, and (d) is a vertical line. Let’s calculate the slopes for all four graphs using the slope formula.

  The slope of the line in figure (a) on page 57 is .

  This tells us that if we wanted to move from one point on the line to another point on the line, all we would have to do is move up three units vertically and one unit to the right horizontally.

  The slope of a line that rises is always positive.

  The slope of the line in figure (b) is .

  The slope of a line that falls is always negative.

  The slope of the line in figure (c) is .

  The slope of a horizontal line is always 0.

  The slope of the line in figure (d) is undefined.

  The slope of a vertical line is undefined.

  SELF-TEST 2:

  Calculate the slopes of the following graphs:

  1.

  2.

  3.

  4.

  ANSWERS:

  1.

  2.

  3.

  4.

  3 Writing the Equation of a Straight Line

  There are two basic formulas we can use to write the equation of a line. The first is y = mx + b, where b is the y-intercept and m is the slope. We can’t use this formula unless we know the y-intercept. Remember, we can tell a point is a y-intercept if its x coordinate is 0. The second formula is y − y1 = m(x − x1 ); we don’t need to know the y-intercept to use this formula.

  Example 5:

  Write the equation of the line with m = 2 that passes through the point (0,5).

  Solution:

  We know the point (0,5) is the y-intercept because x is 0; b = 5. If we substitute b = 5 and m = 2 into the formula y = mx + b we get the equation of the line, which is y = 2x

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  Improving Teacher Knowledge in K-12 Schooling

  Perspectives on STEM Learning

  • Xiaoxia A. Newton(Author)
  • 2018(Publication Date)
  • Palgrave Macmillan

    (Publisher)

  First, all respondents define slope formulaically as rise over run using two points on the line (or symbolically as). Defining slope in this way in our view creates several conceptual difficulties for learners. To begin with, how do we know any two points will work? Secondly, what does it really mean slope is change in y with unit change in x (where in the formula did unit come into play)? Thirdly, what is the connection between the algebraic expression of slope and its graphical/geometric representation? In contrast, the level 4 response defines the slope by directly using the graph of the linear equation and shows on the graph what it means slope is the rise of y over 1 unit of x and that this definition of slope is independent of the point chosen. Once the definition of slope is complete, the response builds on the definition and scaffolds students through a purposeful and coherent process to derive the key ideas that slope of a line can be calculated using any two distinct points, for example P and S, on the line and that we can calculate the slope of a line by dividing the length of the vertical line segment by the length of the horizontal line segment of ΔPST (see Fig. 4.1). This purposefulness brings mathematical closure to students. Fig. 4.1 Calculating slope of a line Second, a majority of respondents took what needs to be proven as given and engaged in circular reasoning. In other words, instead of proving that the slope of a line can be calculated using any two distinct points on the line, they started with the premise that the slope is constant and therefore the formula definition of slope using the two pairs of points shown on the graph is the same. A few considered using a good pedagogical practice of exploration (i.e., try a few points and observe); however, they conflated demonstration through a few examples with mathematical proof

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