Quadratic Transformations into Equations

Quadratic Transformations into Equations

Learning Target: Given the transformation, write the vertex form of the quadratic equation.

We have learned how to analyze parts of a quadratic function to identify the components, such as:

• y = -4(x - 3)² + 2

• Vertex: (3, 2)

• Axis of symmetry: x = 3

• Maximum: y = 2

• Transformations: x-axis reflection, vertical stretch x4, shift right 3, shift up 2

Today, we are going to work backwards! When given the transformation, we will need to find the equation.

We will learn through trial and error...

Remember that y = a(x - h)² + k is the in-general equation, where

• (h, k) are the coordinates of the vertex

• axis of symmetry is x = h

• maximum/minimum is y = k

Remember that y = a(x - h)² + k is the in-general equation, where

• reflection when there is a (-) sign in front

• vertical stretch when a is >1

• vertical compression when 0 < a < 1

• vertical shift up/down when you +/- k

• horizontal shift left/right when you +/- h

Multiple Choice

Which transformation maps the graph of
f(x) = x² to the graph of g(x) = (x + 4)²?

• a reflection across the line x = -4
• a reflection across the line y = -4
• a translation shifting f(x) 4 units to the left
• a translation shifting f(x) 4 units to the right

Multiple Choice

What steps transform the graph y = x² to y = (x - 4)²

• Shifted down 4 units

• Shifted left 4 units

• Shifted right 4 units

• Shifted up 4 units

Multiple Choice

What steps transform the graph y = x² to y = x² + 8

• shifted up 8 units

• shifted down 8 units

• shifted left 8 units

• shifted right 8 units

Multiple Choice

Identify the vertex of f(x) = 5(x+6)² - 7

• (5,6)

• (0,0)

• (6,-7)

• (-6,-7)

Multiple Choice

What steps transform the graph y = x² to y = -1/2(x-4)² + 1?

• Compressed by 1/2, shifted 4 units right and 1 unit up

• Reflected across x-axis, compressed by 1/2, shifted 4 units right and 1 unit up

• Reflected across x-axis, stretched by 1/2, shifted 4 units left and 1 unit up

• Stretched by 1/2, shifted 4 units right and 1 unit down

Multiple Choice

Let the graph of g be a translation 7 units right and 12 units up of the graph of f(x) = x². Write a rule for g.

• g(x) = (x + 7)² + 12

• g(x) = (x + 12)² + 7

• g(x) = (x - 7)² - 12

• g(x) = (x - 7)² + 12

Multiple Choice

Let the graph of g be a translation 7 units right and a reflection in the x-axis of the graph of f(x) = x². Write a rule for g.

• g(x) = -(x - 7)²

• g(x) = (x - 7)²

• g(x) = -(x + 7)²

• g(x) = (x + 7)²

Multiple Choice

Let the graph of g be a vertical stretch by a factor of 5 of the graph of f(x) = x². Write a rule for g.

• g(x) = 5x²

• g(x) = -5x²

• g(x) = x² + 5

• g(x) = (x + 5)²

Multiple Choice

Let the graph of g be a translation 2 units down and 4 units left of the graph of f(x) = x². Write a rule for g.

• g(x) = (x - 4)² - 2

• g(x) = (x + 4)² - 2

• g(x) = (x - 4)² + 2

• g(x) = (x + 4)² + 2