linear transformations, Quadratic Transformations, Quadratic Transformations

f(x) = -2(x - 3)² - 3

The parabola has a maximum with a vertex of (3, -3). The parabola is narrow because it changes faster than the parent quadratic function.

f(x) = -2(x + 3)² - 3

The parabola opens down; it's narrowed towards the line of symmetry; the vertex is (-3, -3).

parabola

a U-shaped graph

y = x²

basic parabola

y=f(1/2x)

horizontal stretch by a factor of 2

y = − (1/5)x²

stretched (made flatter by a factor of 1/5) and reflected

vertex

the lowest or highest point on a parabola

axis of symmetry

-b/2a

f(x) = 3(x + 2)² - 1

The parabola has a minimum with a vertex of (-2, -1). The parabola was dilated by narrowing; there was no reflection; and it was translated left 2 units and down 1 unit from the parent quadratic function.

f(x) = x² - 3

The parabola has a minimum with a vertex of (0, -3). The parabola was not dilated or reflected from the parent quadratic function; does not shift horizontally; only translated down 3 units.

f(x) is the parent functions. What transformation happens in f(x) = 1/2(x - 1)² ?

The parabola narrows and shifts to the right one. Does not transform vertically.

f(x) = -(x + 3)²

The parabola opens down with no dilation (does not widen or narrow). It is translated to the left 3 units. Does not translate up or down.

f(x) = (-1/4)(x + 1)² - 4

The parabola opens down; it widens; the parent quadratic function is translated to the left 1 unit and down 4 units.

f(x) = -(x + 4)² - 3

The parabola opens down; the graph changes at the same speed as the parent quadratic function (does not narrow or widen); the vertex is (-4, -3); translates to the left 4 units and down 3 units.

f(x) = .75 (x - 1)² - 3

The parabola opens up; it widens (the speed of the graph is slower than the parent function); the parent quadratic function is translated to the right 1 unit and down 3 units.

f(x) = 4(x - 1)² - 3

The parabola opens up; it's narrowed towards the line of symmetry (it's speed is faster than the parent function); translates left one, down 3.

f(x) = (x)² - 1

The parabola translates down 1, does not shift horizontally.

f(x) is the parent function. What transformation happens in f(x) = (x + 2)² ?

The parabola translates left 2 units only.

f(x) is the parent function. What transformation happens in f(x) + 2 ?

The parabola translates up 2 units only.

y=f(2x)

horizontal compression by a factor of 1/2

y = x² − 3

moved down 3

y = (x+2)²

moved left 2

y = (x − 3)²

moved right 3

y=-f(x)

reflected over the x-axis

y=-f(x+4)

reflected over the x-axis and shifted left 4 units

y=f(-x)

reflected over the y-axis

y = −x²

reflected over x-axis (opens downward)

y=f(x) -4

shifted down 4 units

y=f(x-4)

shifted to right 4 units

y=f(x+4)

shifted to the left 4 units

y=f(x)+4

shifted up 4 units

y = −5x²

shrunk (made more narrow by a factor of 5) and reflected

x-intercept

the point on a graph where the line crosses the x-axis

y-intercept

the point on a graph where the line crosses the y-axis

y=1/2f(x)

vertical compression by a factor of 1/2

y=2f(x)

vertical stretch by a factor of 2

y = x² + 3

moved up 3

y=2f(x-4)

vertically stretched by a factor of 2 and shifted right 4 units