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Algebra 1 Notes

Learn how to identify, graph, and solve linear equations using intercepts and rate of change. Understand the meaning of intercepts and rate of change in real-world situations.

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Algebra 1 Notes

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  1. Algebra 1 Notes Chapter 3

  2. 3-1 Notes for Algebra 1 Graphing Linear Equations

  3. Linear Equation An equation that forms a line when it graphed. Standard Form for a linear equation is C is called a constant (or a number) Ax and By are variable terms.

  4. Example 1 pg. 155 Identify Linear Equations Determine whether each equation is a linear equation. Write the equation in standard form. 1.) 2.)

  5. Example 1 pg. 155 Identify Linear Equations Determine whether each equation is a linear equation. Write the equation in standard form. 1.) Not linear 2.) Linear;

  6. x-intercept The point where the line crosses the x-axis

  7. y-intercept The point where the line crosses the y-axis.

  8. Example 2 pg. 156 Find intercepts from a graph Find the x- and y-intercepts of the segment graphed below.

  9. Example 2 pg. 156 Find intercepts from a graph Find the x- and y-intercepts of the segment graphed below. x-intercept is 4 y-intercept is 200

  10. Example 3 pg. 157 Find intercepts from a table. ANALYZE TABLES A box of peanuts is poured into bags at a rate of 4 ounces per second. The table shows the function relating the weight of peanuts in the box and the time in seconds the peanuts have been pouring out of the box. 1.) Find the x- and y-intercepts of the graph of the function. 2.) Describe what the intercepts mean in terms of this situation.

  11. Example 3 pg. 157 Find intercepts from a table. ANALYZE TABLES A box of peanuts is poured into bags at a rate of 4 ounces per second. The table shows the function relating the weight of peanuts in the box and the time in seconds the peanuts have been pouring out of the box. 1.) Find the x- and y-intercepts of the graph of the function. x-int.=500, y-int.=2000 2.) Describe what the intercepts mean in terms of this situation. x-int.: 0 oz. after 500 s. y-int.: 2000 oz. before pouring began

  12. Example 4 pg. 157 Graph by using intercepts Graph by using the x- and y-intercepts.

  13. Example 4 pg. 157 Graph by using intercepts Graph by using the x- and y-intercepts.

  14. Example 5 pg. 158 Graph by making a table Graph

  15. Example 5 pg. 158 Graph by making a table Graph

  16. 3-1 pg. 159 13-49o, 42, 50, 51-55o, 66-75)(x3)

  17. 3-2 Notes for Algebra 1 Solving Linear Equations by Graphing.

  18. Linear Function A function for which the graph is a line. The simplest linear function is a called a parent function. A Family of Graphs is a group of graphs with one or more similar characteristics.

  19. Solution/root Any value that makes an equation true. Zeros—values of x for which Zeros, Roots, solutions and x-intercepts are terms that represent the same thing.

  20. Example 1 pg. 164 solve an Equation with one root. 1.) 2.)

  21. Example 1 pg. 164 solve an Equation with one root. 1.) -6 2.) -3

  22. Example 2 pg. 164 Solve an Equation with no solution 1.) 2.)

  23. Example 2 pg. 164 Solve an Equation with no solution 1.) 2.)

  24. Example 3 pg. 165 Estimate by Graphing FUNDRAISING Kendra’s class is selling greeting cards to raise money for new soccer equipment. They paid $115 for the cards, and they are selling each card for $1.75. The function represents their profit y for selling x greeting cards. Find the zero of this function. Describe what this value means in this context.

  25. Example 3 pg. 165 Estimate by Graphing FUNDRAISING Kendra’s class is selling greeting cards to raise money for new soccer equipment. They paid $115 for the cards, and they are selling each card for $1.75. The function represents their profit y for selling x greeting cards. Find the zero of this function. Describe what this value means in this context. ; they must sell 66 cards to make a profit.

  26. 3-2 pg. 166 11-43o, 51-66(x3)

  27. 3-3 Notes for Algebra 1 Rate of Change and Slope

  28. Rate of Change A ratio that describes how much one quantity changes with respect to a change in another quantity. x—is the independent variable. y—is the dependent variable. A positive rate of change indicates an increase over time. A negative rate of change indicates that a quantity is decreasing.

  29. Example 1 pg. 172 Find the Rate of Change DRIVING TIME Use the table to find the rate of change. Then explain its meaning.

  30. Example 1 pg. 172 Find the Rate of Change DRIVING TIME Use the table to find the rate of change. Then explain its meaning. ; this means the car is traveling at a rate of 38 miles per hour.

  31. Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 1.) Find the rate of change for 2002-2004 9 and 2004-2006. 7 2.) Explain the meaning of the rate of change 5 in each case. 0 2002 2004 2006 3.) How are the different rates of change shown on the graph. 12.1 8.9 7.0

  32. Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 1.) Find the rate of change for 2002-2004 9 and 2004-2006. 7 5 0 2002 2004 2006 950,000/yr 1,600,000/yr 12.1 8.9 7.0

  33. Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 9 7 2.) Explain the meaning of the rate of change 5 in each case. 0 2002 2004 2006 For 2002-2004 there was an annual increase of 950,000 passports issued. Between 2004-2006, there was an average yearly increase of 1,600,000 passports issued. 12.1 8.9 7.0

  34. Example 2 pg. 173 Compare Rates of Change TRAVEL The graph shows the number of 13 U.S. passports issued in 2002, 2004 and 2006 11 9 7 5 0 2002 2004 2006 3.) How are the different rates of change shown on the graph. There is a greater vertical change for 2004-2006 than for 2002-2004. Therefore, the section of the graph for 2004-2006 is steeper. 12.1 8.9 7.0

  35. Slope (m) The slope of a non-vertical line is the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run) ,

  36. Example 4 pg. 175 Positive, Negative and Zero slopes Find the slope of the line that passes through each pair of points. 1.) 2.) 3.) and

  37. Example 4 pg. 175 Positive, Negative and Zero slopes Find the slope of the line that passes through each pair of points. 1.) 2.) 3.) and

  38. Example 5 pg. 176 Undefined Slope Find the slope of the line that passes through and .

  39. Example 5 pg. 176 Undefined Slope Find the slope of the line that passes through and . Undefined

  40. Slope Summary Positive Slope– goes uphill from left to right Negative Slope– goes downhill from left to right Slope of 0– horizontal line (y = #) Undefined Slope– vertical line (x = #)

  41. Example 6 pg. 176 Find Coordinates given the slope. Find the value of r so that the line through and has a slope of .

  42. Example 6 pg. 176 Find Coordinates given the slope. Find the value of r so that the line through and has a slope of .

  43. 3-3 pg. 177 15-39o, 43-45, 54-66(x3)

  44. 3-4 Notes for Algebra 1 Direct Variation

  45. Direct Variation Is described as an equation of the form , where . This is a constant rate of change. k is the constant of variation/proportionality

  46. Example 1 pg. 182 Slope and Constant of Variation. Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points. 1.) 2.) y = -4x y = 2x (0, 0) (1, 2) (0, 0) 0000 (1, -4)

  47. Example 1 pg. 182 Slope and Constant of Variation. Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points. 1.) 2.) Constant of variation: 2; slope:2 Constant of Variation: -4; slope: -4 y = -4x y = 2x (0, 0) (1, 2) (0, 0) 0000 (1, -4)

  48. Example 2 pg. 183 Graph a Direct Variation Graph

  49. Example 2 pg. 183 Graph a Direct Variation Graph

  50. Example 3 pg. 183 Write and solve a Direct Variation Equation Suppose y varies directly as x and when . 1.) Write a direct variation equation that relates x and y. 2.) Use the direct variation equation to find x when .

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