Math  /  Algebra

Question42. Find the domain of each function: (A) f(x)=2x5x2x6f(x)=\frac{2 x-5}{x^{2}-x-6} (B) g(x)=3x5xg(x)=\frac{3 x}{\sqrt{5-x}}
In Problems 57-59, find the equation of any horizontal asymptote.
57. f(x)=5x+4x23x+1f(x)=\frac{5 x+4}{x^{2}-3 x+1}
58. f(x)=3x2+2x14x25x+3f(x)=\frac{3 x^{2}+2 x-1}{4 x^{2}-5 x+3}
53. Explain how the graph of m(x)=x4m(x)=-|x-4| is related to the graph of y=xy=|x|.
54. Explain how the graph of g(x)=0.3x3+3g(x)=0.3 x^{3}+3 is related to the graph of y=x3y=x^{3}.
19. Complete the square and find the standard form for the quadratic function f(x)=x2+4xf(x)=-x^{2}+4 x

Then write a brief verbal description of the relationship between the graph of ff and the graph of y=x2y=x^{2}.

Studdy Solution
Complete the square for f(x)=x2+4x f(x) = -x^2 + 4x .
First, factor out the negative sign:
f(x)=(x24x) f(x) = -(x^2 - 4x)
To complete the square, take half of the coefficient of x x , square it, and add and subtract it inside the parentheses:
f(x)=(x24x+44) f(x) = -(x^2 - 4x + 4 - 4) f(x)=((x2)24) f(x) = -((x-2)^2 - 4) f(x)=(x2)2+4 f(x) = -(x-2)^2 + 4
The standard form is f(x)=(x2)2+4 f(x) = -(x-2)^2 + 4 .
Describe its transformation relative to y=x2 y = x^2 :
The graph of y=x2 y = x^2 is reflected over the x-axis, shifted 2 units to the right, and 4 units up.

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