Algebra Practice: Graphs, Lines, and Circles

Flashcards covering coordinate geometry concepts including distance and midpoint formulas, symmetry, linear equations, and circle equations based on Chapter 1 practice problems.

What is the distance $d(P_1, P_2)$ between the points $P_1 = (5, 5)$ and $P_2 = (5, 2)$?

$$3$$

What is the distance $d(P_1, P_2)$ between the points $P_1 = (1, 3)$ and $P_2 = (-2, -1)$?

$$5$$

How do you determine if points such as $(-9, -1)$, $(-4, -1)$, and $(-4, 7)$ form a right triangle?

By checking if the square of the longest side equals the sum of the squares of the other two sides (Pythagorean Theorem) or if two sides are perpendicular.

What are the values of $k$ such that the points $(-5, 5)$ and $(k, 0)$ are $\sqrt{29}$ units apart?

$$-3, -7$$

A middle school baseball field is a square with side lengths of $65\,ft$. To the nearest foot, what is the distance from home plate to second base along the diagonal?

$$92\,ft$$

If a motorcycle travels north at $20\,mph$ and a car travels east at $48\,mph$ from the same intersection, what is the expression for their distance apart in miles at the end of $t$ hours?

$$52t\,miles$$

What is the midpoint of the line segment joining points $P_1 = (1, 5)$ and $P_2 = (9, 3)$?

$$(5, 4)$$

If $(-3, 1)$ is an endpoint of a line segment and $(2, 4)$ is its midpoint, what is the other endpoint?

$$(7, 7)$$

In which quadrant does the point $(-2, 6)$ lie?

Quadrant II

On which axis does the point $(0, -4)$ lie?

$$y\text{-axis}$$

What are the intercepts for the graph of the equation $y = x + 6$?

$(-6, 0)$ and $(0, 6)$

If a graph is symmetric with respect to the $y\text{-axis}$ and contains the point $(5, -6)$, what other point must be on the graph?

$$(-5, -6)$$

If a graph is symmetric with respect to the origin and contains the point $(-4, 7)$, what other point must be on the graph?

$$(4, -7)$$

Interpret a slope of $m = \frac{1}{11}$ in the context of coordinate changes.

For every $11\text{-unit}$ increase in $x$, $y$ will increase by $1\,unit$.

What is the slope of the line containing the points $(-5, 4)$ and $(-2, -6)$?

$$-\frac{10}{3}$$

What is the equation for a vertical line containing the point $(-5, 6)$?

$$x = -5$$

What is the equation of a horizontal line containing the point $(7, -3)$?

$y = -3$$

Find the equation of the line in general form that contains the points $(4, 0)$ and $(0, -5)$.

$$5x - 4y = 20$$

What is the slope and $y\text{-intercept}$ of the line defined by the equation $12x - 8y = 96$?

$$\text{slope} = \frac{3}{2}; y\text{-intercept} = -12$$

Find the equation of the line parallel to $y = -4x$ that contains the point $(2, 3)$.

$$y = -4x + 11$$

Find the equation of the line perpendicular to $y = \frac{1}{8}x + 8$ that contains the point $(4, -4)$.

$$y = -8x + 28$$

What is the standard form of the equation of a circle with radius $r = 12$ and center $(h, k) = (-6, 3)$?

$$(x + 6)^2 + (y - 3)^2 = 144$$

Find the center $(h, k)$ and radius $r$ of the circle with the equation $(x - 8)^2 + (y - 3)^2 = 121$.

$$(h, k) = (8, 3); r = 11$$

Find the center $(h, k)$ and radius $r$ of the circle $x^2 + y^2 - 12x - 16y = -96$.

$$(h, k) = (6, 8); r = 2$$

If a black bear transmitter has a range of $21\,miles$ and the research station is the origin $(0, 0)$, what equation represents the transmitter's maximum range?

$$x^2 + y^2 = 441$$