Chapter 5

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the cube root of \(x\). When \(x=27,\) then \(y=5 .\) Find \(y\) when \(x=125 .\)

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

For the following exercises, find the slant asymptote of the functions. $$ f(x)=\frac{6 x^{3}-5 x}{3 x^{2}+4} $$

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ x^{4}-2 x^{3}-7 x^{2}+8 x+12=0 $$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(x^{4}-8 x^{3}+24 x^{2}-32 x+16\right) \div(x-2) $$

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

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(1,0),(x, y)=(0,1) $$

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as \(x\) and \(z\). When \(x=4\) and \(z=2\), then \(y=16\). Find \(y\) when \(x=3\) and \(z=3\).

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

For the following exercises, find the slant asymptote of the functions. $$ f(x)=\frac{x^{2}+5 x+4}{x-1} $$

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ x^{4}+2 x^{3}-9 x^{2}-2 x+8=0 $$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(x^{4}+5 x^{3}-3 x^{2}-13 x+10\right) \div(x+5) $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=(2 x+1)^{3}\left(9 x^{2}-6 x+1\right) $$

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

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 given transformation to graph the function. Note the vertical and horizontal asymptotes. $$ p(x)=\frac{2 x-3}{x+4} $$

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ 4 x^{4}+4 x^{3}-25 x^{2}-x+6=0 $$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(x^{4}-12 x^{3}+54 x^{2}-108 x+81\right) \div(x-3) $$

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 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 the Rational Zero Theorem to find the real solution(s) to each equation. $$ 2 x^{4}-3 x^{3}-15 x^{2}+32 x-12=0 $$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(4 x^{4}-2 x^{3}-4 x+2\right) \div(2 x-1) $$

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. $$ f(x)=x^{2}+4 x, \quad x \geq-2 $$

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 the Rational Zero Theorem to find the real solution(s) to each equation. $$ x^{4}+2 x^{3}-4 x^{2}-10 x-5=0 $$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(4 x^{4}+2 x^{3}-4 x^{2}+2 x+2\right) \div(2 x+1) $$

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. $$ f(x)=x^{2}-6 x+1, \quad x \geq 3 $$

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 the Rational Zero Theorem to find the real solution(s) to each equation. $$ 4 x^{3}-3 x+1=0 $$

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. $$ x-2, \quad 4 x^{3}-3 x^{2}-8 x+ 4 $$

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, 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 the Rational Zero Theorem to find the real solution(s) to each equation. $$ 8 x^{4}+26 x^{3}+39 x^{2}+26 x+6 $$

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. $$ x-2, \quad 3 x^{4}-6 x^{3}-5 x+10 $$

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 \(t=5\).