SOLUTION: Prove Analytically that f(x)=x^3-4x is odd. Prove Analytically the f(x)=x^4+2x^2+5 is even. Write the equation for a function that has a graph with the shape of y=square root

Algebra ->  Functions -> SOLUTION: Prove Analytically that f(x)=x^3-4x is odd. Prove Analytically the f(x)=x^4+2x^2+5 is even. Write the equation for a function that has a graph with the shape of y=square root      Log On


   

Question 54753: Prove Analytically that f(x)=x^3-4x is odd.
Prove Analytically the f(x)=x^4+2x^2+5 is even.
Write the equation for a function that has a graph with the shape of y=square root of x, but is reflected in the y-axis and shifted up 12.
Write the equation for a function that has a graph of y=x^2, but is vertically shrunk by a factor of 0.2 and shifted left 15 units.
Answer by funmath(2933) About Me  (Show Source):
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Prove Analytically that f(x)=x^3-4x is odd.
Substitute -x in for x, if f(-x)=-f(x) then the function is odd.
f%28-x%29=%28-x%29%5E3-4%28-x%29 remember (-1)^(odd#)=-1
f%28-x%29=-x%5E3%2B4x
Therefore f(-x)=-f(x), so the function is odd.
:
Prove Analytically the f(x)=x^4+2x^2+5 is even.
Substitute -x in for x, if f(-x)=f(x) then the function is even.
f%28-x%29=%28-x%29%5E4%2B3%28-x%29%5E2%2B5 remember (-1)^(even#)=1
f%28-x%29=x%5E4%2B3x%5E2%2B5
Therefore f(-x)=f(x), so the function is even.
:
Write the equation for a function that has a graph with the shape of y=square root of x, but is reflected in the y-axis and shifted up 12.
f(-x) results in a reflection about the y-axis.
f(x)+k recults in a vertical shift k units.
y=sqrt%28x%29 Looks like:
graph%28300%2C200%2C-10%2C10%2C-10%2C10%2Csqrt%28x%29%29
y=sqrt%28-x%29 reflects about y-axis.
y=sqrt%28-x%29%2B12 reflects it about y-axis and shiftes the graph up 12 units.
It looks like:
graph%28300%2C200%2C-10%2C10%2C-5%2C15%2Csqrt%28-x%29%2B12%29
:
Write the equation for a function that has a graph of y=x^2, but is vertically shrunk by a factor of 0.2 and shifted left 15 units.
af(x) results in a vertical shrink if -1 f(x+k) results in a horizontal shift left of k units.
y=x%5E2 Looks like:
graph%28300%2C200%2C-10%2C10%2C-10%2C10%2Cx%5E2%29
y=.2x%5E2 results in a vertical shink of a factor of .2
y=.2%28x%2B15%29%5E2 results in the shrink and the left shift of 15 units.
It looks like:
graph%28300%2C200%2C-20%2C5%2C-10%2C10%2C.2%28x%2B15%29%5E2%29
Happy Calculating!!!