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Transforming Linear Equations

Transforming Linear Equations. Linear Equations. Family of functions – a set of functions whose graphs have basic characteristics in common. For example, all linear functions form a family because all of their graphs are the same (they are lines).

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Transforming Linear Equations

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  1. Transforming Linear Equations

  2. Linear Equations • Family of functions – a set of functions whose graphs have basic characteristics in common. For example, all linear functions form a family because all of their graphs are the same (they are lines). • Parent Function – the most basic function in a family. For linear functions, the parent function is f(x)= x.

  3. Linear Equations • Transformation – A change in position or size of a figure. • Three types of transformations • Translation (slide) • Rotation (turn) • Reflection (flip)

  4. Translations • When we “slide” the parent function f(x) = x, it will move the function up or down on the y-axis. • This changes the y-intercept (b). • The slopes (m) will stay the same.

  5. Notice all the lines on this graph are parallel. The slopes are the same but the y-intercepts are different.

  6. The graphs of g(x) = x + 3, h(x) = x – 2, and k(x) = x – 4, are vertical translations of the graph of the parent function, f(x) = x. A translation is a type of transformation that moves every point the same distance in the same direction. You can think of a translation as a “slide.”

  7. Reflections • When we reflect (flip) a transformation across a line it produces a mirror image. • When the slope (m) is multiplied by -1 the graph is reflected across the y-axis.

  8. The diagram shows the reflection of the graph of f(x) = 2x across the y-axis, producing the graph of g(x) = –2x. A reflection is a transformation across a line that produces a mirror image. You can think of a reflection as a “flip” over a line.

  9. f(x) g(x) To find g(x), multiply the value of m by–1. In f(x) = x + 2, m = . g(x) = – x + 2 Graph . Then reflect the graph of f(x) across the y-axis. Write a function g(x) to describe the new graph.

  10. Rotation • Rotation – a transformation about a point. • You can think of a rotation as a “turn”. • The y-intercepts are the same, but the slopes are different. • When the slope is changed it will cause a rotation about the point that is the y-intercept changing the line’s steepness.

  11. The graphs of g(x) = 3x, h(x) = 5x, and k(x) = are rotations of the graph f(x) = x. A rotation is a transformation about a point. You can think of a rotation as a “turn.” The y-intercepts are the same, but the slopes are different.

  12. What change in the parent function causes… • A translation? • A rotation? • A reflection? Changing the y-intercept (b), slope is the same Changing the slope, y-intercept stays the same Multiplying the slope by -1, y-intercept stays the same

  13. Graph f(x) = x and g(x) = 2x – 3. Then describe the transformations from the graph of f(x) to the graph of g(x). Find transformations of f(x) = xthat will result in g(x) = 2x – 3: h(x) = 2x • Multiply f(x) by 2 to get h(x) = 2x. This rotates the graph about (0, 0) and makes it parallel to g(x). f(x) = x • Then subtract 3 from h(x) to get g(x) = 2x –3. This translates the graph 3 units down. g(x) = 2x – 3 The transformations are a rotation and a translation.

  14. A florist charges $25 for a vase plus $4.50 for each flower. The total charge for the vase and flowers is given by the function f(x) = 4.50x + 25. How will the graph change if the vase’s cost is raised to $35? If the charge per flower is lowered to $3.00? Total Cost f(x) = 4.50x + 25 is graphed in blue. If the vase’s price is raised to $35, the new function is f(g) = 4.50x + 35. The original graph will be translated 10 units up.

  15. A florist charges $25 for a vase plus $4.50 for each flower. The total charge for the vase and flowers is given by the function f(x) = 4.50x + 25. How will the graph change if the vase’s cost is raised to $35? If the charge per flower is lowered to $3.00? Total Cost If the charge per flower is lowered to $3.00. The new function is h(x) = 3.00x + 25. The original graph will be rotated clockwise about (0, 25) and become less steep.

  16. f(x) = x, g(x) = 2x Try these… Describe the transformation from the graph of f(x) to the graph of g(x). 1.f(x) = 4x, g(x) = x 2. 3. 4. rotation (less steep) f(x) = x –1, g(x) = x + 6 translated 7 units up Rotation (steeper) f(x) = 5x, g(x) = –5x Reflection

  17. Try these… Part II 5.f(x) = x, g(x) = x – 4 6. translation 4 units down f(x) = –3x, g(x) = –x + 1 rotation (less steep), translation 1 unit up 7. A cashier gets a $50 bonus for working on a holiday plus $9/h. The total holiday salary is given by the function f(x) = 9x + 50. How will the graph change if the bonus is raised to $75? if the hourly rate is raised to $12/h? translation 25 units up; rotated (steeper)

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