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 = x² + 3
moved up 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=2f(x-4)
vertically stretched by a factor of 2 and shifted right 4 units