Final Communicate

Generally, explain what each variable a, b, c, and d does in g(x)=af(b(x+c))+d.

In the transformation of f(x) to g(x)=af(b(x+c))+d, the variable a determines whether the function is vertically stretched or compressed, and whether there is a reflection over the x-axis. The variable b determines whether the function is horizontally stretched or compressed, and whether there is a reflection over the y-axis. The variable c determines where the function will shift horizontally. The variable d determines where the function will shift vertically.

g(x)=3(2^-x+5)-1

The parent function is f(x)=2^x. The transformations needed to get g(x)=3(2^-x+5)-1 from the parent function are: a reflection over the y-axis, a shift right 5 units, a vertical stretch by a factor of 3, and a shift down 1 unit.

g(x)=-2sin(4πx-π)+3

The parent function is f(x)=sin(x). The transformations needed are a reflection over the x-axis, a phase shift 1/4 unit right, shrink the period from 4π to 1/2, stretch the amplitude from 1 to 2, and a shift up 3 units.

Describe how to find csc(-17π/6) using the Unit Circle.

Draw θ=-17π/6 in standard position on the Unit Circle. Find the corresponding coordinates. Use the sine value of the point and apply the reciprocal identity csc(θ)=1/sin(θ).

Give an example of a rational function with horizontal asymptote y=3/4. Justify.

An example is f(x)=(3x+1)/(4x-2). The horizontal asymptote is y=3/4 because the degrees of the numerator and denominator are equal and the ratio of leading coefficients is 3/4.

Give an example of a rational function with no horizontal asymptote. Justify.

An example is f(x)=x²/(x+1). There is no horizontal asymptote because the degree of the numerator is greater than the degree of the denominator.

Give an example of a rational function with horizontal asymptote y=0. Justify.

An example is f(x)=1/x. The horizontal asymptote is y=0 because the degree of the numerator is less than the degree of the denominator.

Give an example of a rational function with a slant asymptote. Justify.

An example is f(x)=(x²+1)/x. There is a slant asymptote because the degree of the numerator is exactly one greater than the degree of the denominator.

Give an example of a rational function with vertical asymptote x=1. Justify.

An example is f(x)=1/(x-1). There is a vertical asymptote at x=1 because the denominator equals zero at x=1.

Are the domains of f(x)=log(x+3) and g(x)=log(x²-2x-15) the same?

The domains are not the same. For f(x), the domain is x>-3. For g(x), the domain is x

Is cos⁻¹(cos(7π/6))=7π/6 correct? Justify.

The student is incorrect. The range of cos⁻¹(x) is [0,π]. Since 7π/6 is not in the range, the expression evaluates to 5π/6, not 7π/6.

Explain the difference between log₃(x-1)=4 and log₃(x-1)=log₃(4).

The first equation is solved by converting to exponential form. The second equation is solved using the one-to-one property of logarithms.

If tan(θ)<0 and sin(θ)<0, what quadrant is θ in?

θ lies in Quadrant IV because sin(θ) is negative in Quadrants III and IV, and tan(θ) is negative in Quadrants II and IV, so the intersection is Quadrant IV.

A point (-4,3) is on the terminal side of θ. A student says cos(θ)=-4. Is the student correct?

The student is incorrect. Cosine is x/r, where r=5, so cos(θ)=-4/5, not -4.

Inverse functions relationship + exponential inverse

Inverse functions undo each other. The inverse of an exponential function is a logarithmic function. The graph is found by reflecting across y=x.

How many solutions does sin(θ)=-1/2 have on [0,2π)?

There are two solutions. Sine is negative in Quadrants III and IV. The reference angle is π/6. The solutions are 7π/6 and 11π/6.