Parent function for any quadratic function will be y = x^{2}
Comparing the given function with this vertex form, we can decide the transformations that we have to do.
y = a(x - h)^{2 }+ k, where
Find the following for the quadratic functions below.
Vertex:
Transformations:
Domain :
Range :
Max/min:
Axis of symmetry:
y-intercept:
Problem 1 :
y = (x + 2)^{2} + 2
Solution:
Vertex:
y = a(x - h)^{2} + k
y = (x + 2)^{2} + 2
Vertex (h, k) = (-2, 2)
Transformations:
Parent function is shifted 2 units left and 2 units up.
Domain:
All real numbers.
x € R
Range:
y ≥ 2
{y € R| y ≥ 2}
Max/min:
Min = 2
Axis of symmetry:
x = h
x = -2
y-intercept:
set x = 0
y = (0 + 2)^{2} + 2
y = 4 + 2
y = 6
y-intercept = (0, 6)
Problem 2 :
y = 3(x + 1)^{2} + 3
Solution:
Vertex:
y = a(x - h)^{2} + k
y = 3(x + 1)^{2} + 3
Vertex (h, k) = (-1, 3)
Transformations:
Parent function is vertically stretched by shifted 1 unit left and 3 units up.
Domain:
All real numbers.
x € R
Range:
y ≥ 3
{y € R| y ≥ 3}
Max/min:
Min = 3
Axis of symmetry:
x = h
x = -1
y-intercept:
set x = 0
y = 3(0 + 1)^{2} + 3
y = 3 + 3
y = 6
y-intercept = (0, 6)
Problem 3 :
y = -(x - 2)^{2} + 7
Solution:
Vertex:
y = a(x - h)^{2} + k
y = -(x - 2)^{2} + 7
Vertex (h, k) = (2, 7)
Transformations:
Parent function is inverted about the x-axis. shifted 2 units right and 7 units up.
Domain:
All real numbers.
x € R
Range:
y ≤ 7
{y € R| y ≤ 7}
Max/min:
Max = 7
Axis of symmetry:
x = h
x = 2
y-intercept:
set x = 0
y = -(0 - 2)^{2} + 7
y = -4 + 7
y = 3
y-intercept = (0, 3)
Problem 4 :
y = -2(x - 3)^{2} + 3
Solution:
Vertex:
y = a(x - h)^{2} + k
y = -2(x - 3)^{2} + 3
Vertex (h, k) = (3, 3)
Transformations:
Parent function is vertically stretched by shifted 3 units right and 3 units up.
Domain:
All real numbers.
x € R
Range:
y ≤ 3
{y € R| y ≤ 3}
Max/min:
Max = 3
Axis of symmetry:
x = h
x = 3
y-intercept:
set x = 0
y = -2(0 - 3)^{2} + 3
y = -18 + 3
y = -15
y-intercept = (0, -15)
Problem 5 :
y = (x + 5)^{2} - 1
Solution:
Vertex:
y = a(x - h)^{2} + k
y = (x + 5)^{2} - 1
Vertex (h, k) = (-5, -1)
Transformations:
Parent function is shifted 5 units left and 1 unit down.
Domain:
All real numbers.
x € R
Range:
y ≥ -1
{y € R| y ≥ -1}
Max/min:
Min = -1
Axis of symmetry:
x = h
x = -5
y-intercept:
set x = 0
y = (0 + 5)^{2} - 1
y = 24
y-intercept = (0, 24)
Problem 6 :
y = 1/2(x + 5)^{2} - 1
Solution:
Vertex:
y = a(x - h)^{2} + k
y = 1/2(x + 5)^{2} - 1
Vertex (h, k) = (-5, -1)
Transformations:
Parent function is vertically stretched by 5 units left and 1 unit down.
Domain:
All real numbers.
x € R
Range:
y ≥ -1
{y € R| y ≥ -1}
Max/min:
Min = -1
Axis of symmetry:
x = h
x = -5
y-intercept:
set x = 0
y = 1/2(0 + 5)2 - 1
y = 11.5
y-intercept = (0, 11.5)
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM