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4.1 – Patterns and Lines

Objectives

  • Explore relationships of direct variation, such as diameter to circumference.
  • Derive the equation of the line from the best-fit line.
  • Construct a linear equation from an English sentence
  • Graph a linear equation from a chart of solutions to the equation.

 

Key Terms

  • Cartesian Coordinate System – An xy-plane formed by a horizontal number line (the x-axis) and a vertical number line (the y-axis) that intersect at the zero point of each.
    • It is also called the xy-plane or coordinate plane.
  • Coordinates – The x- and y-values that describe the location of a point in an xy-plane.
  • x-coordinate – The coordinate of a point that gives its distance to the right or left of the origin on a coordinate plane.
    • The x-coordinate is the first number in the ordered pair that gives the point’s location.
  • y-coordinate – The coordinate of a point that gives its distance above or below the origin on a coordinate plane.
    • The y-coordinate is the second number in the ordered pair that gives the point’s location.
  • Direct variation – A relationship in which one variable (quantity) is a constant multiple of the other variable (quantity).
    • You can say that y varies directly with x and write y = kx, where k is a constant.
    • The relationship of direct variation is plotted on a graph as a straight line.
    • The graph passes through the origin.

 

km traveled (y) varies directly with Hours driven (x). k = 60 (the constant speed).

Alg1B 4.1 Direct Variation

 

  • Equation of the Line – Equation that shows the relationship between the x-value and y-value of every point on the line.
  • Graphs of Linear Equations – The set of all points on a line whose coordinates can be substituted into an equation and keep that equation true.
  •  Line – A straight path that continues forever in both directions.
    • The symbol \overleftrightarrow{PQ} represents a line that contains points P and Q.
  • Line of Best Fit – A line drawn as near as possible to the points in a scatterplot.
    • The best-fit line helps you see the relationship shown by the scatterplot.
  • Linear Equation – Any equation whose graph is a straight line.
    • A linear equation can be written in the form y = mx + b.
  • Origin – The point where the axes intersect in a Cartesian coordinate system.
    • The coordinates of the origin are (0, 0).

 

Review

  • Circumference and Diameter
    • Circumference is the distance around a circle.
    • Diameter is the distance across the center of a circle.
    • The relationship between the circumference and the diameter is a constant value: π (pi), which is about 3.14 units.
      • Constant values are values that never change.
      • The relationship between circumference and diameter can be written as: y = 3.14x, where 3.14 is the constant.
      • English: The circumference of the wheel depends on its diameter.
      • Algebra: The circumference of the wheel “is a fuction of” its diameter.

Alg1B 4.1 Circumference

 

  • Multiples
    • A multiple is the product of any two whole numbers.
    • Examples
      • 21 is a multiple of 3 because 3 7 = 21.
      • 100 is a multiple of 10 because 10 10 = 100.

 

  • Kilometers
    • Units of length in the metric system of measurement.
    • There are 1000 meters in 1 kilometer.

 


Notes

  • You can use constant values to find multiples of numbers.
    • Constant = 4, so, 1(4)=4, 2(4)=8, 3(4)=12, 4(4)=16, and so on!
    • 4, 8, 12, and 16 are all multiples of 4!

 

  • Linear Equations
    • A linear equation can be written in the form y = mx + b.
    • y = 12x is a linear equation.
    • So, in a linear equation, b can equal zero.

 

  • Direct Variation is a Linear Equation: y=kx
    • You can rewrite the equation as k=\frac{y}{x} by dividing x on both sides.
    • k is a constant and will never change.
    • y=kx is a literal equation, as there are 2 variables: y and x.

 

  • Constant Values (k) or (m) represent a linear equation’s slope
    • The larger the value, the steeper the slope (line goes almost straight up from left to right)
    • y = 10x

Alg1B 4.1 Yis10x

 

    • The smaller the value, the less steep the slope (the line hardly goes up at all from left to right)
    • y = 0.1x

Alg1B 4.1 Yis0.1x

 


  • Positive and Negative Constants
    • Lines that go up from left to right have a positive constant (k) value.
    • Lines that go down from left to right have a negative constant (k) value.

 


  • To find the constant (k) value, when given a graph of a line with points plotted:
    • Change y=kx to k=y÷x by dividing x on both sides.
    • Then, substitute in the x and y point values for ONE point and solve for k.
    • Lastly, reduce the fraction and write the equation in the form y=kx, substituting the answer for k.

 

  • Example 1
    • Points: (3,6) and (1,2) are both plotted on the line in the graph below.
      • You can use EITHER point, as they connect to form a straight line.  All straight lines have a constant value (k).
    • For (3,6):  If x=3 and y=6, then use the formula: k=\frac{y}{x} with the values substituted: k=\frac{6}{3}.
      • k=2 when you reduce this fraction, so the equation is: y=2x
    • For (1,2):  If x=1 and y=2, then use the formula: k=\frac{y}{x} with the values substituted: k=\frac{2}{1}.
      • k=2 when  you reduce the fraction, so the equation is: y=2x once again!

Alg1B 4.1 Write Equation

 

  • Example 2
    • What is the equation of a line that includes the point (2,-6) and goes through the origin (0,0) of the xy-plane?
    • k=\frac{y}{x} so, k=\frac{-6}{2}
    • k = -3
    • The equation will be y = -3x
    • k is negative, which means the line goes down from left to right on the graph.

 


  • Example: Pizza
    • You measure straight across your pizza.
    • It has a diameter of 22 centimeters.
    • We can use the linear equation y = 3.14x to find the circumference of the pizza!

Alg1B 4.1 Pizza

 


  •  Example: Changing Feet to Inches
    • There are 12 inches in a foot, so y = 12x
      • y: inches
      • x: feet
      • 12: constant

Alg1B 4.1 Inches Feet


  • Example: for the line y=\frac{2}{5}x, if the y-coordinate is 20, what is the x-coordinate?
    • Substitute 20 in for y and solve: 20=\frac{2}{5}x
      • 1. Multiply by 5 on both sides: 100 = 2x
      • 2. Divide by 2 on both sides: 50 = x.

 


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