For questions 1 and 2, use integer values of from –3 to 3 to graph the equation.
1) y = –2x2 + 3
2) y = one-third |x| –2
Find the horizontal change and the vertical change for the translation
3) p(4, –4) right arrow (–4, 7).
4) The point c(3, –1) is translated to the left 4 units and up 1 unit. a. Write the rule for this translation.
b. What are the coordinates of the image point?
5) How many lines of symmetry does the figure below have? If there are no lines of symmetry, write none
6) The vertices of ΔABC are A(2, –5), B(–3, 5), and C(3, –3). The triangle is reflected over the x-axis. Use arrow notation to describe the original triangle and its reflection.
7) The point c (x, y) is reflected over the x-axis. Write a translation rule to describe the original point and its reflection.
#1,#2 use any online grapher, such as wolframalpha.com
#3 well, what do you have to add to 4 to get to -4?
(4,-4) + (-8,?) = (-4,7)
#4. Put the words into algebra:
T: (x,y) -> (x-4,y+1)
...
#5 no figure available
#6 to reflect in the x-axis, change the signs of the y-coordinates: (x,y) -> (x,-y)
#7: see #6
This should get you well on your way, assuming you don't just want someone else to do it all for you...
danke!
1) To graph the equation y = -2x^2 + 3, we can create a table of values by substituting integer values for x:
x | y
-------
-3 | -15
-2 | -7
-1 | 1
0 | 3
1 | 1
2 | -1
3 | -3
Plotting these points on a graph and connecting them will give us the graph of the equation.
2) To graph the equation y = one-third |x| - 2, we can again create a table of values:
x | y
-------
-3 | -3
-2 | -2.67
-1 | -2.33
0 | -2
1 | -1.67
2 | -1.33
3 | -1
Plotting these points on a graph and connecting them will give us the graph of the equation.
3) The horizontal change for the translation p(4, -4) --> (-4, 7) is 4 - (-4) = 8.
The vertical change for the translation p(4, -4) --> (-4, 7) is 7 - (-4) = 11.
4) a. The rule for the translation of point c(3, -1) to the left 4 units and up 1 unit is: (x, y) --> (x - 4, y + 1).
b. To find the coordinates of the image point, we apply the translation rule:
c(3, -1) --> (3 - 4, -1 + 1) --> (-1, 0). Therefore, the coordinates of the image point are (-1, 0).
5) The figure is not described, so we cannot determine the lines of symmetry.
6) The original triangle ΔABC can be described with arrow notation as A(2, -5) --> B(-3, 5) --> C(3, -3).
After reflecting it over the x-axis, the new triangle would be A'(2, 5) --> B'(-3, -5) --> C'(3, 3).
7) The original point c(x, y) reflected over the x-axis can be described with the translation rule as: (x, y) --> (x, -y).
1) To graph the equation y = -2x^2 + 3, you can substitute different values for x and calculate the corresponding values for y.
For example, when x = -3, y = -2(-3)^2 + 3 = -2(9) + 3 = -18 + 3 = -15. So one point on the graph is (-3, -15).
Similarly, for x = -2, -1, 0, 1, 2, and 3, you can calculate the corresponding values of y and plot those points on a graph. Then, connect the points to form a smooth curve.
2) To graph the equation y = one-third|x| - 2, you can use a similar approach.
For x = -3, y = 1/3|-3| - 2 = 1/3(3) - 2 = 1 - 2 = -1. So one point on the graph is (-3, -1).
Repeat this process for x = -2, -1, 0, 1, 2, and 3 to find the corresponding points and plot them on the graph. Connect the points to form the graph.
3) To find the horizontal and vertical change for the translation from p(4, -4) to (-4, 7), you can subtract the x-coordinates and y-coordinates of the new point from the original point.
Horizontal Change: (-4) - 4 = -8
Vertical Change: 7 - (-4) = 11
So, the horizontal change is -8 and the vertical change is 11.
4) a) To write the rule for the translation of the point c(3, -1) to the left 4 units and up 1 unit, you need to subtract 4 from the x-coordinate and add 1 to the y-coordinate. The rule can be written as:
(x, y) -> (x - 4, y + 1)
b) Applying the translation rule, the new coordinates of the image point will be:
3 - 4 = -1 (x-coordinate)
-1 + 1 = 0 (y-coordinate)
So, the image point is (-1, 0).
5) To determine the number of lines of symmetry in a figure, you need to analyze its symmetry. If the figure can be folded along a line so that one half is an exact mirror image of the other half, then it has a line of symmetry. Counting the lines that meet this criterion, you can identify the number of lines of symmetry.
If the figure cannot be folded along any line to create a mirror image, it has no lines of symmetry.
6) To describe the original triangle and its reflection over the x-axis using arrow notation, you can specify the coordinates of each vertex.
Original Triangle ΔABC:
A(2, -5) -> (-2, -5)
B(-3, 5) -> (-3, -5)
C(3, -3) -> (3, 3)
Reflection of ΔABC over the x-axis:
A'(-2, -5) -> (-2, 5)
B'(-3, -5) -> (-3, 5)
C'(3, 3) -> (3, -3)
Note that the x-coordinates remain the same, but the y-coordinates are negated.
7) To write a translation rule that describes the reflection of point c(x, y) over the x-axis, you need to negate the y-coordinate. The translation rule can be written as:
(x, y) -> (x, -y)
So, the original point is reflected over the x-axis by negating its y-coordinate.