Answer:
[tex]y=\dfrac{1}{2}(x+5)^2-3[/tex]
Step-by-step explanation:
Translations
For [tex]a > 0[/tex]
[tex]f(x+a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units left}[/tex]
[tex]f(x)-a \implies f(x) \: \textsf{translated}\:a\:\textsf{units down}[/tex]
[tex]y=a\:f(x) \implies f(x) \: \textsf{stretched parallel to the y-axis (vertically) by a factor of}\:a[/tex]
[tex]y=-f(x) \implies f(x) \: \textsf{reflected in the} \: x \textsf{-axis}[/tex]
Parent function
[tex]y=-x^2[/tex]
Reflected in the x-axis
[tex]-f(x) \implies y=-(-x^2)\implies y=x^2[/tex]
Compressed vertically by a factor of 1/2
Multiply the whole function by the given scale factor:
[tex]\dfrac{1}{2}f(x)\implies y=\dfrac{1}{2}x^2[/tex]
Translated 3 units down
Subtract 3 from the whole function:
[tex]f(x)-3 \implies y=\dfrac{1}{2}x^2-3[/tex]
Translated 5 units left
Add 5 to the variable of the function:
[tex]f(x+5) \implies y=\dfrac{1}{2}(x+5)^2-3[/tex]
To sketch the parabola
Vertex = [tex](-5, -3)[/tex]
Axis of symmetry: [tex]x = -5[/tex]
Plot points:
[tex]x=-9 \implies \dfrac{1}{2}(-9+5)^2-3=5 \implies (-9,5)[/tex]
[tex]x=-7 \implies \dfrac{1}{2}(-7+5)^2-3=-1 \implies (-7,-1)[/tex]
[tex]x=-3 \implies \dfrac{1}{2}(-3+5)^2-3=-1 \implies (-3,-1)[/tex]
[tex]x=-1 \implies \dfrac{1}{2}(-1+5)^2-3=5 \implies (-1,5)[/tex]
