Graphs

If the point (2, 5) is shifted 3 units right and 2 units down, what are its new coordinates?

If the point (-1, 6) is shifted 2 units left and 4 units up, what are its new coordinates?

The medians of a triangle are the line segments from each vertex to the midpoint of the opposite side (see the figure). Find the lengths of the medians of the triangle with vertices at A = (0, 0), B = (6, 0), and C = (4, 4).

An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are (0, 4) and (0, 0), find the third vertex. How many of these triangles are possible?

In Problems 101–104, find the length of each side of the triangle determined by the three points P1 , P2 , and P3 . State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length.)

$P_1=\left(2,1\right),P_2=\left(-4,1\right),P_3=\left(-4,-3\right)$

In Problems 101–104, find the length of each side of the triangle determined by the three points P1 , P2 , and P3 . State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length.)

In Problems 101–104, find the length of each side of the triangle determined by the three points P1 , P2 , and P3 . State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length.)

$P_1=\left(-2,-1\right),P_2=\left(0,7\right),P_3=\left(3,2\right)$

In Problems 101–104, find the length of each side of the triangle determined by the three points P1 , P2 , and P3 . State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length.)

$P_1=\left(7,2\right),P_2=\left(-4,0\right),P_3=\left(4,6\right)$

Baseball A major league baseball “diamond” is actually a square, 90 feet on a side (see the figure). What is the distance directly from home plate to second base (the diagonal of the square)?

Little League Baseball The layout of a Little League playing field is a square, 60 feet on a side. How far is it directly from home plate to second base (the diagonal of the square)?

Baseball Refer to Problem 105. Overlay a rectangular coordinate system on a major league baseball diamond so that the origin is at home plate, the positive x-axis lies in 

the direction from home plate to first base, and the positive y-axis lies in the direction from home plate to third base.

(a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement.

(b) If the right fielder is located at (310, 15), how far is it from there to second base?

(c) If the center fielder is located at (300, 300), how far is it from there to third base?

Little League Baseball Refer to Problem 106. Overlay a rectangular coordinate system on a Little League baseball diamond so that the origin is at home plate, the positive

x-axis lies in the direction from home plate to first base, and the positive y-axis lies in the direction from home plate to third base.

(a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement.

(b) If the right fielder is located at (180, 20), how far is it from there to second base?

(c) If the center fielder is located at (220, 220), how far is it from there to third base?

Distance between Moving Objects A Ford Focus and a Mack truck leave an intersection at the same time. The Focus heads east at an average speed of 30 miles per hour,

while the truck heads south at an average speed of 40 miles per hour. Find an expression for their distance apart d (in miles) at the end of t hours.

Distance of a Moving Object from a Fixed Point A hot-air balloon, headed due east at an average speed of 15 miles per hour at a constant altitude of 100 feet, passes over an

intersection (see the figure). Find an expression for its distance d (measured in feet) from the intersection t seconds later.

Drafting Error When a draftsman draws three lines that are to intersect at one point, the lines may not intersect as intended and subsequently will form an error triangle. If this

error triangle is long and thin, one estimate for the location of the desired point is the midpoint of the shortest side. The figure shows one such error triangle.

(a) Find an estimate for the desired intersection point.

(b) Find the length of the median for the midpoint found in part (a). See Problem 99.

Net Sales The figure illustrates how net sales of Wal-Mart Stores, Inc., have grown from 2006 through 2010. Use the midpoint formula to estimate the net sales of Wal-Mart

Stores, Inc., in 2008. How does your result compare to the reported value of $374 billion?

Poverty Threshold Poverty thresholds are determined by the U.S. Census Bureau. A poverty threshold represents the minimum annual household income for a family not

to be considered poor. In 2001, the poverty threshold for a family of four with two children under the age of 18 years was \(17,960. In 2011, the poverty threshold for a family of four with two children under the age of 18 years was \)22,350. Assuming poverty thresholds increase in a straight-line fashion, use the midpoint formula to estimate the poverty threshold of a family of four with two children under the age of 18 in 2006. How does your result compare to the actual poverty threshold in 2006 of $20,444?

Completing a Line Segment Plot the points A = (-1, 8) and M = (2, 3) in the $xy$-plane. If M is the midpoint of a line segment AB, find the coordinates of B.

(a) Graph $y=\sqrt{x^2}$ $y=x,y=\left|x\right|$, and $y={\left(\sqrt{x}\right)}^2$

noting which graphs are the same.

(b) Explain why the graphs of$y=\sqrt{x^2}$ and $y=\left|x\right|$ are the same.

(c) Explain why the graphs of y = x and $y={\left(\sqrt{x}\right)}^2$ are not the same.

(d) Explain why the graphs of $y=\sqrt{x^2}$and y = x are not the same.

Make up an equation satisfied by the ordered pairs $\left(2,0\right),\left(4,0\right),\left(0,1\right)$ Compare your equation with a friend’s equation. Comment on any similarities.

Draw a graph that contains the points $\left(-2,-1\right),\left(0,1\right),\left(1,3\right),\left(3,5\right)$. Compare your graph with those of other students. Are most of the graphs almost straight lines? How

many are “curved”? Discuss the various ways that these points might be connected.

Explain what is meant by a complete graph.

Write a paragraph that describes a Cartesian plane. Then write a second paragraph that describes how to plot points in the Cartesian plane. Your paragraphs should include the terms “coordinate axes,” “ordered pair,” “coordinates,” “plot,” “x-coordinate,” and “y-coordinate.”

Solve: $2\left(x+3\right)-1=-7$

Solve:

$x^2-4x-12=0$

The points, if any, at which a graph crosses or touches the

coordinate axes are called ___________.

The x-intercepts of the graph of an equation are those x-values for which ___________.

If for every point (x, y) on the graph of an equation the point (-x, y) is also on the graph, then the graph is symmetric with respect to the ___________.

If the graph of an equation is symmetric with respect to the y-axis and -4 is an x-intercept of this graph, then ________ is also an x-intercept.

If the graph of an equation is symmetric with respect to the origin and (3, -4) is a point on the graph, then ________ is also a point on the graph.

True or False :

To find the y-intercepts of the graph of an equation, let x = 0 and solve for y.

True or False :

 The y-coordinate of a point at which the

graph crosses or touches the x-axis is an x-intercept.

True or False : 

If a graph is symmetric with respect to the x-axis, then it cannot be symmetric with respect to the y-axis.

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

y = x + 2

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$y=x-6$.

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$y=2x+8$.

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$y=3x-9$.

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$y=x^2-1$.

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$y=x^2-9$.

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$y=-x^2+4.$

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$y=-x^2+1$.

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$2x+3y=6$.

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$5x+2y=10$.

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$9x^2+4y=36$.

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

$4x^2+y=4$.

Plot each point. Then plot the point that is symmetric to it with respect to -

(a) the x-axis

(b) the y-axis

(c) the origin

Point  (3, 4)

Plot each point. Then plot the point that is symmetric to it with respect to -

(a) the x-axis

(b) the y-axis

(c) the origin

Point (5, 3)

Plot each point. Then plot the point that is symmetric to it with respect to -

(a) the x-axis

(b) the y-axis

(c) the origin

Point (-2, 1)

Plot each point. Then plot the point that is symmetric to it with respect to -

(a) the x-axis

(b) the y-axis

(c) the origin

Point (4, -2)

Plot each point. Then plot the point that is symmetric to it with respect to -

(a) the x-axis

(b) the y-axis

(c) the origin

Point (5, -2)