Question

Guys, can you please help me with The Question #47 of The Quadratic Relations for me, please? :)

Answer 1

`y=(1)/(2)(x+5)^2-3`

Explanation:

Translations

For
`a > 0`

`f(x+a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units left}`

`f(x)-a \implies f(x) \: \textsf{translated}\:a\:\textsf{units down}`

`y=a\:f(x) \implies f(x) \: \textsf{stretched parallel to the y-axis (vertically) by a factor of}\:a`

`y=-f(x) \implies f(x) \: \textsf{reflected in the} \: x \textsf{-axis}`

Parent function

`y=-x^2`

Reflected in the x-axis

`-f(x) \implies y=-(-x^2)\implies y=x^2`

Compressed vertically by a factor of 1/2

Multiply the whole function by the given scale factor:

`(1)/(2)f(x)\implies y=(1)/(2)x^2`

Translated 3 units down

Subtract 3 from the whole function:

`f(x)-3 \implies y=(1)/(2)x^2-3`

Translated 5 units left

Add 5 to the variable of the function:

`f(x+5) \implies y=(1)/(2)(x+5)^2-3`

To sketch the parabola

Vertex =
`(-5, -3)`

Axis of symmetry:
`x = -5`

Plot points:

`x=-9 \implies (1)/(2)(-9+5)^2-3=5 \implies (-9,5)`

`x=-7 \implies (1)/(2)(-7+5)^2-3=-1 \implies (-7,-1)`

`x=-3 \implies (1)/(2)(-3+5)^2-3=-1 \implies (-3,-1)`

`x=-1 \implies (1)/(2)(-1+5)^2-3=5 \implies (-1,5)`