Sketch $(1, −2, 3)$ and $\left(-\frac{1}{3},\frac{2}{3},-1\right)$. Why do these vectors point in opposite directions?

Find a point-slope form of the line satisfying the conditions. Use the first point given for $\left(x_1, y_1\right)$. Then convert the equation to slope-intercept form, and write the formula for a function $f$ whose graph is the line.

$x$-intercept $(-2,0), y$-intercept $(0,5)$

Graph the parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range.

$$x=y^2+2 y-8$$

Identify the condition in the following diagnoses.
Bullous keratopathy, left eye
main term: __________

Plot the point whose polar coordinates are given and find its rectangular coordinates. Give answers in simplest radical form.

$$\left( - 3,120 ^ { \circ } \right)$$

Simplify. All variables represent nonnegative numbers.

$$\frac { 1 } { 2 } \sqrt { \frac { 1 } { 25 } }$$

Graph the parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range.

$$x=(y-3)^2$$

Let x(t) and y(t) be cubic Bezier curves with control points $\left\{\mathbf{p}_{0}, \mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3}\right\}$ and $\left\{\mathbf{p}_{3}, \mathbf{p}_{4}, \mathbf{p}_{5}, \mathbf{p}_{6}\right\}$, respectively, so that x(t) and y(t) are joined at $\mathbf{p}_{3}$. The following questions refer to the curve consisting of x(t) followed by y(t). For simplicity, assume that the curve is in $\mathbb{R}^{2}$. a. What condition on the control points will guarantee that the curve has $C^{1}$ continuity at $\mathbf{p}_{3}$? Justify your answer. b. What happens when x'(1) and y'(0) are both the zero vector?

Graph the parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range.

$$x=-2(y+3)^2$$

Find the volume of the solid formed by revolving the region bounded by the graphs of the given equations about the x-axis. x = 6 cos θ, y = 6 sin θ

In this exercise, you will evaluate the composite function

Let $f(x)=\sqrt{x}$ and $g(x)=x^2-1$. Evaluate the expression given below.

$$g(f(0))$$

Find the area of the surface generated by revolving the curve $x=2+\cos t, y=1+\sin t$, for $0 \leq t \leq 2 \pi$ about the x-axis.

Find all vertical asymptotes. $f(x)=\frac{x^{2}+1}{x^{3}+3 x^{2}+2 x}$

A solid S is generated by revolving the region between the x-axis and the curve y=

$$\sqrt { \sin x } ( 0 \leq x \leq \pi )$$

about the x-axis. An integral expression for the volume of S is ___.