A baker has 10 cups of sugar to make cookies. Each batch calls for 1 1/3 cups of sugar.
2 answers:
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A baker has 10 cups of sugar to make cookies. Each batch calls for 1 1/3 cups of sugar.
is a mixed number, we could write it in fraction form
.
cups of sugar required for = 1 batch
Hence 1 cup of sugar required for =
Hence 1 cup of sugar required for
So, 10 cups of sugar required for =
So, 10 cups of sugar required for =
could be written as
or we could write
So, he could make batches of cookies. : Answer
Hope it will help :)
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Answer:
4) The linear equation which could represent its image is y = x + 8 ⇒ (2)
5) The equation of the image of the line is y = x - 3 ⇒ (4)
Step-by-step explanation:
Dilation does not change the slope of a line but changes the y-intercept
Dilation of a line segment is longer or shorter by ratio that equal to the scale factor of dilation
4)
The slope-intercept form of the linear equation is y = m x + b, where m is the slope of the line and b its y-intercept
Let us put the given equation in the slope-intercept form to find its slope
∵ The line -3x + 4y = 8 is transformed by a dilation centered at
the origin
- Add 3x to both sides
∴ 4y = 3x + 8
- Divide both sides by 4
∴ y = x + 2 ⇒ the equation of the line before dilation
- By comparing it with the form above, then m = , so the
equation after dilation has the same slope
∵ The slope of the line before dilation is
∴ The slope of the line after dilation is
∵ The equation that has the slope is y =
∴ The equation of the image of the line is y = x + 8
The linear equation which could represent its image is y = x + 8
5)
∵ The equation of the line is y = x - 4
∵ The scale factor of dilation is
- Dilation dose not change the slope of the line
∴ The slope of the image after dilation is
- Dilation changes the y-intercept, to find it multiply the scale
factor of dilation by the y-intercept of the line before dilation
∵ The y-intercept of the line is (0 , -4)
- Multiply it by
∵ × -4 = -3
∴ The y-intercept of the image is (0 , -3)
∴ The equation of the image of the line is y = x - 3
Even function:
A function is said to be even if its graph is symmetric with respect to the , that is:
Odd function:
A function is said to be odd if its graph is symmetric with respect to the origin, that is:
So let's analyze each question for each type of functions using examples of polynomial functions. Thus:
FOR EVEN FUNCTIONS:
1. When becomes
1.1 Effects on the y-intercept
We need to find out the effects on the y-intercept when shifting the function into:
We know that the graph intersects the y-axis when , therefore:
So:
So the y-intercept of is one unit less than the y-intercept of
1.2. Effects on the regions where the graph is increasing and decreasing
Given that you are shifting the graph downward on the y-axis, there is no any effect on the intervals of the domain. The function increases and decreases in the same intervals of
1.3 The end behavior when the following changes are made.
The function is shifted one unit downward, so each point of has the same x-coordinate but the output is one unit less than the output of . Thus, each point will be sketched as:
FOR ODD FUNCTIONS:
2. When becomes
2.1 Effects on the y-intercept
In this case happens the same as in the previous case. The new y-intercept is one unit less. So the graph is shifted one unit downward again.
An example is shown in Figure 1. The graph in blue is the function:
and the function in red is:
So you can see that:
2.2. Effects on the regions where the graph is increasing and decreasing
The effects are the same just as in the previous case. So the new function increases and decreases in the same intervals of
In Figure 1 you can see that both functions increase at:
and decrease at:
2.3 The end behavior when the following changes are made.
It happens the same, the output is one unit less than the output of . So, you can write the points just as they were written before.
So you can realize this concept by taking a point with the same x-coordinate of both graphs in Figure 1.
FOR EVEN FUNCTIONS:
3. When becomes
3.1 Effects on the y-intercept
We need to find out the effects on the y-intercept when shifting the function into:
As we know, the graph intersects the y-axis when , therefore:
And:
So the new y-intercept is the negative of the previous intercept shifted one unit upward.
3.2. Effects on the regions where the graph is increasing and decreasing
In the intervals when the function increases, the function decreases. On the other hand, in the intervals when the function decreases, the function increases.
3.3 The end behavior when the following changes are made.
Each point of the function has the same x-coordinate just as the function and the y-coordinate is the negative of the previous coordinate shifted one unit upward, that is:
FOR ODD FUNCTIONS:
4. When becomes
4.1 Effects on the y-intercept
In this case happens the same as in the previous case. The new y-intercept is the negative of the previous intercept shifted one unit upward.
4.2. Effects on the regions where the graph is increasing and decreasing
In this case it happens the same. So in the intervals when the function increases, the function decreases. On the other hand, in the intervals when the function decreases, the function increases.
4.3 The end behavior when the following changes are made.
Similarly, each point of the function has the same x-coordinate just as the function and the y-coordinate is the negative of the previous coordinate shifted one unit upward.
