2.4.6 Translating, reflecting, stretching graphs, functions

• What is the mathematical expression for translation?

  • f(x) + a
  • f(x+a)
• What are the three types of graph transformations shown in the image?

  Translation, Stretch, Reflection
• What are the three main types of graph transformations?

  Translation, stretch, reflection
• What determines the direction and magnitude of the graph's movement in a translation?

  The value of 'a'
• What is the equation of the parabola on the right side of the image?

  $y =$$x^2$
• What is the formula for the reflection of a function f(x) in the x-axis?

  • The reflected function is -f(x)
• How do reflections affect the graph of a function?

  • Reflection in the x-axis multiplies the y-coordinates by -1
  • Reflection in the y-axis multiplies the x-coordinates by -1
• What is the coefficient in the function f(x) = 2x + 3?


  2
• What are the two main types of graph transformations shown in the image?

  • Translations
  • Reflections
• How do translations affect the graph of a function?

  • Translation by vector (a, 0) adds a to the x-coordinate
  • Translation by vector (0, a) adds a to the y-coordinate
• What is the formula for a translation of a function f(x) by the vector (a, 0)?


  f(x - a)
• If a function f(x) is stretched horizontally by a factor of 3, what is the new function?

  f(x/3)
• How do the mathematical expressions for translation, stretch, and reflection differ?

  Translation adds/subtracts a constant, stretch multiplies/divides the input, and reflection negates the input
• What is the formula for a translation of a function f(x) by the vector (0, a)?


  f(x) + a
• If a function f(x) is reflected over the y-axis, what is the new function?

  f(-x)
• How do the equations of the two parabolas differ?

  • Left parabola: $y =$$(2x)^2$
  • Right parabola: $y =$$x^2$

  The left parabola has a coefficient of 2 in front of x, making it narrower and more compressed horizontally compared to the right parabola.
• What is the region between the two parabolas called?

  Horizontal compression
• The variable in a function is always represented by 'x'.

  False
• What is a translation in the context of graphs?

  Moves every point by same amount
• What type of graph transformation is explored in this material?

  Reflections across axes
• What happens to the graph when $f(x)$ is transformed to f(kx)</latex> with $0 < k < 1$?


  Graph becomes wider
• How do the properties of the two parabolas, such as their vertex, axis of symmetry, and concavity, compare?

  The parabolas have the same vertex at the origin (0,0), the same axis of symmetry along the y-axis, and opposite concavity (one is concave up, the other is concave down)
• What is the equation of the parabola on the right side of the graph?

  $y =$$-(x^2)$
• What is the equation of the reflection of $y =$$x^{2}$ across the x-axis?


  $y =$$- x^{2}$
• What is the equation of the parabola on the left side of the image?

  $y =$$(2x)^2$
• How can you use the graph to determine the domain and range of the two parabolic functions?

  • Domain: The x-values from -10 to 10
  • Range: The y-values from -10 to 10 for both parabolas
• What is the relationship between the two parabolas shown in the graph?

  • The two parabolas are reflections of each other over the x-axis
  • One parabola is the positive square function (y = x^2)
  • The other parabola is the negative square function (y = -(x^2))
• The transformation $f(kx)$ stretches or compresses the graph vertically.


  False
• How could you use this graph to solve a quadratic equation?

  • To solve a quadratic equation of the form y = x^2 or y = -(x^2), you can use the graph to find the x-intercepts
  • The x-intercepts represent the solutions to the quadratic equation
• If a function f(x) is reflected in the x-axis, how does the y-coordinate change?

  The y-coordinates are multiplied by -1
• What is the name of the graph transformation shown in the image?

  Graph transformation
• What is the mathematical expression for reflection?

  • -f(x)
  • f(-x)
• What does changing 'k' in a trigonometric function $f(x) =$$A \sin(kx)$ do to the graph?


  Stretches or compresses horizontally
• What is the standard notation used to represent a function?

  f(x)
• What happens to the graph when $f(x)$ is transformed to $f(kx)$ with $k > 1$?


  Graph becomes narrower
• How do you identify transformations of a function?

  By its equation
• What is the mathematical expression for stretch?

  • af(x)
  • f(ax)
• What transformation does $f(x) +$$a$ perform on the graph of $f(x)$?


  Shifts it up or down
• What is the role of constants in a function?

  Fixed numbers
• The variable in a function can only be 'x', 'y', or 'z'.

  False