Chapter 5

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts. $$ f(x)=x^{2}-2 x $$

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as \(x, z,\) and \(w .\) When \(x=2, z=1,\) and \(w=12\), then \(y=72\). Find \(y\) when \(x=1, z=2\), and \(w=3\).

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$f(x)=x^{3}+3$$

For the following exercises, use the Rational Zero Theorem to find all real zeros. \(4 x^{4}+4 x^{3}-25 x^{2}-x+6=0\)

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes. The reciprocal function shifted up two units.

For the following exercises, use synthetic division to find the quotient. $$ \left(x^{4}-12 x^{3}+54 x^{2}-108 x+81\right) \div(x-3) $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=(3 x+2)^{5}\left(x^{2}-10 x+25\right) $$

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts. $$ f(x)=x^{2}-6 x-1 $$

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as \(x\) and the square of \(z\). When \(x=2\) and \(z=4,\) then \(y=144 .\) Find \(y\) when \(x=4\) and \(z=5\).

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$f(x)=1-x^{3}$$

For the following exercises, use the Rational Zero Theorem to find all real zeros. \(2 x^{4}-3 x^{3}-15 x^{2}+32 x-12=0\)

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes. The reciprocal function shifted down one unit and left three units.

For the following exercises, use synthetic division to find the quotient. $$ \left(4 x^{4}-2 x^{3}-4 x+2\right) \div(2 x-1) $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=x\left(4 x^{2}-12 x+9\right)\left(x^{2}+8 x+16\right) $$

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts. $$ f(x)=x^{2}-5 x-6 $$

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as the square of \(x\) and the square root of \(z\). When \(x=2\) and \(z=9\), then \(y=24\). Find \(y\) when \(x=3\) and \(z=25\).

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$f(x)=x^{2}+4 x, x \geq-2$$

For the following exercises, use the Rational Zero Theorem to find all real zeros. \(x^{4}+2 x^{3}-4 x^{2}-10 x-5=0\)

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes. The reciprocal squared function shifted to the right 2 units.

For the following exercises, use synthetic division to find the quotient. $$ \left(4 x^{4}+2 x^{3}-4 x^{2}+2 x+2\right) \div(2 x+1) $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=x^{6}-x^{5}-2 x^{4} $$

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts. $$ f(x)=x^{2}-7 x+3 $$

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as \(x\) and \(z\) and inversely as \(w .\) When \(x=5, z=2,\) and \(w=20,\) then \(y=4 .\) Find \(y\) when \(x=3\) and \(z=8,\) and \(w=48\).

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$f(x)=x^{2}-6 x+1, x \geq 3$$

For the following exercises, use the Rational Zero Theorem to find all real zeros. \(4 x^{3}-3 x+1=0\)

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes. The reciprocal squared function shifted down 2 units and right 1 unit.

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x-2,4 x^{3}-3 x^{2}-8 x+4 $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=3 x^{4}+6 x^{3}+3 x^{2} $$

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts. $$ f(x)=-2 x^{2}+5 x-8 $$

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as the square of \(x\) and the cube of \(z\) and inversely as the square root of \(w\). When \(x=2\), \(z=2\), and \(w=64\), then \(y=12\). Find \(y\) when \(x=1\), \(z=3,\) and \(w=4\).

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$f(x)=\frac{2}{x}$$

For the following exercises, use the Rational Zero Theorem to find all real zeros. \(8 x^{4}+26 x^{3}+39 x^{2}+26 x+6\)

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ p(x)=\frac{2 x-3}{x+4} $$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x-2,3 x^{4}-6 x^{3}-5 x+10 $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=4 x^{5}-12 x^{4}+9 x^{3} $$

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts. $$ f(x)=4 x^{2}-12 x-3 $$

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as the square of \(x\) and of \(z\) and inversely as the square root of \(w\) and of \(t\). When \(x=2, z=3\), \(w=16,\) and \(t=3,\) then \(y=1 .\) Find \(y\) when \(x=3, z=2, w=36,\) and \(t=5\).

For the following exercises, find all complex solutions (real and non-real). \(x^{3}+x^{2}+x+1=0\)

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ q(x)=\frac{x-5}{3 x-1} $$

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=\frac{1}{x^{2}}, x \geq 0 $$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x+3,-4 x^{3}+5 x^{2}+8 $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=2 x^{4}\left(x^{3}-4 x^{2}+4 x\right) $$

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies directly with the square of \(x\) and when \(x=2\), \(y=3\).

For the following exercises, find all complex solutions (real and non-real). \(x^{3}-8 x^{2}+25 x-26=0\)

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ s(x)=\frac{4}{(x-2)^{2}} $$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x-2,4 x^{4}-15 x^{2}-4 $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=4 x^{4}\left(9 x^{4}-12 x^{3}+4 x^{2}\right) $$

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies directly as the cube of \(x\) and when \(x=2, y=4\).

For the following exercises, use a graph to help determine the domain of the functions. $$f(x)=\sqrt{\frac{(x+2)(x-3)}{x-1}}$$

For the following exercises, find all complex solutions (real and non-real). \(x^{3}+13 x^{2}+57 x+85=0\)