Coordinate Geometry

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Give the domain of the function

$f(x) =x+ \sqrt{x+ 7}$

$\left \{ x| x \ge -7 \right \}$

The set of all real numbers

$\left \{ x| x \ne -7 \right \}$

$\left \{ x| x \ne 7 \right \}$

$\left \{ x| x \ge 7 \right \}$

Explanation

The square root of a real number is defined only for nonnegative radicands; therefore, the domain of is exactly those values for which the radicand is nonnegative. Solve the inequality:

$x+ 7 \ge 0$

$x+ 7 - 7 \ge 0 - 7$

$x \ge - 7$

The domain of is $\left \{ x| x \ge -7 \right \}$ .

In which quadrant does the complex number lie?

Explanation

If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:

We are essentially doing the same as plotting the point on a set of Cartesian axes. We move units right in the x direction, and units down in the y direction, which puts us in the fourth quadrant, or in terms of Roman numerals:

The point on the coordinate plane with coordinates lies

in Quadrant II.

in Quadrant III.

in Quadrant IV.

in Quadrant I.

on an axis.

Explanation

On the coordinate plane, a point with a negative -coordinate and a positive -coordinate lies in the upper left quadrant - Quadrant II.

Give the -coordinate(s) of the -intercept(s) of the graph of the function

$f(x) = \frac{1}{2} x- \frac{7}{2}$

$\frac{1}{2}$

$- \frac{7}{2}$

$\frac{1}{7}$

The graph of has no -intercept.

Explanation

The -intercept(s) of the graph of are the point(s) at which it intersects the -axis. The -coordinate of each is 0; their -coordinate(s) are those value(s) of for which , so set up, and solve for , the equation:

$\frac{1}{2} x- \frac{7}{2} =0$

Add $\frac{7}{2}$ to both sides:

$\frac{1}{2} x- \frac{7}{2} + \frac{7}{2} =0+ \frac{7}{2}$

$\frac{1}{2}x =\frac{7}{2}$

Multiply both sides by 2:

$2 \cdot \frac{1}{2}x =2 \cdot \frac{7}{2}$

the correct choice.

Define $g (x) = \frac{1}{\sqrt[3]{x -27}}$

What is the natural domain of ?

$(-\infty , 27)\cup (27, \infty )$

$(27, \infty )$

$(3, \infty )$

$(-\infty , - 27)\cup (27, \infty )$

$(-\infty , - 3)\cup (3, \infty )$

Explanation

The radical in and of itself does not restrict the domain, since every real number has a real cube root. However, since the expression $\sqrt[3]{x -27}$ is in a denominator, it cannot be equal to zero, so the domain excludes the value(s) for which

$\sqrt[3]{x -27} = 0$

$\left (\sqrt[3]{x -27} \right )^{3}= 0^{3}$

27 is the only number excluded from the domain.

$Find\ the\ vertex\ for\ the\ function: f(x)=4x^{2}-16x+11.$

Explanation

$Use \frac{-b}{2a}\ to\ calculate\ the\ x-value\ of\ the\ vertex.$

$\small Given\ the\ function\ f(x)=4x^{2}-16x+11\ is\ in\ standard\ form\ y=ax^{2}+bx+c$

$\small We\ can\ see\ that\ a=4, b=-16\ and\ c=11$

$\small Substitute\ these\ values\ in\ for\ x=\frac{-b}{2a}\ to\ get\ x=\frac{16}{8}=2$

$\small Now\ substitute\ x=2\ into\ the\ function\ f(x)\ to\ get:$

$\small f(2)=4(2)^{2}-16(2)+11=-5$

$\small The\ coordinates\ of\ the\ vertex\ are\ (2,-5)$

What is the domain of $y = 4 - x^{2}$ ?

all real numbers

$x \leq 4$

$x \geq 4$

$x \leq 0$

Explanation

The domain of the function specifies the values that can take. Here, $4-x^{2}$ is defined for every value of , so the domain is all real numbers.

The point on the coordinate plane with coordinates $\left ( 100, 0 \right )$ lies

on an axis.

in Quadrant IV.

in Quadrant III.

in Quadrant I.

in Quadrant II.

Explanation

On the coordinate plane, a point with 0 as one of its coordinates lies on an axis.

What is the domain of $y = 4 - x^{2}$ ?

all real numbers

$x \leq 4$

$x \geq 4$

$x \leq 0$

Explanation

The domain of the function specifies the values that can take. Here, $4-x^{2}$ is defined for every value of , so the domain is all real numbers.

What is the slope of the line that passes through the points and ?

Explanation

The slope of a line is sometimes referred to as "rise over run." This is because the formula for slope is the change in y-value (rise) divided by the change in x-value (run). Therefore, if you are given two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$ , the slope of their line can be found using the following formula: $slope=(y_{1}-y_{2})/(x_{1}-x_{2})$

This gives us .

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