All right, let's take a moment to review what we've learned. Linear equations in general are functions that graph to form a straight line and are typically written as y = mx + b, where m is the slope and b is the y-intercept. Horizontal lines go left and right and are in the form of y = b, where b represents the y-intercept, while vertical lines go up and down and are in the form of x = a where a represents the shared x-coordinate of all points. All you need to do is remember these.
Horizontal And Vertical Lines Algebra 1 Homework Answers
When we graph lines, we typically begin with a point and then use the slope to determine the line. There are, however, some special exceptions. These are called horizontal and vertical lines.
We can still graph them with ease once we understand what they are and how they work! Continue on to learn more including the slope of a horizontal line, the slope of a vertical line, and what these special lines look like.
The straight lines can be horizontal, vertical or inclined. When the lines are horizontal, the vertical coordinate of all the points through which it passes are equal. When the lines are vertical, the horizontal coordinate of all the points through which it passes are equal. To draw a line we need at least two points.
Please bear with me. I am trying to help my daughter with her Algebra 1 homework. We are asked to describe the transformation of function f to function g as follows:$$f(x) = x$$$$g(x) = 2x+3$$The provided answer states that $g(x)=2x+3$ can be re-written as $$g(x)=2f(x)+3$$ and is therefore a vertical stretch by a factor of 2 (plus a vertical translation up by 3 units). Well and good. However, $g(x)=2x+3$ can also be re-written as $$g(x)=f(2x)+3$$ and be described as a horizontal shrink by a factor of 1/2. But even though this horizontal shrink gives exactly the same graph as the vertical stretch, it is not mentioned as a possible correct answer. I understand that the order of transformations is important and can give completely different graphs if you mess up the order, but this is not the case here. There is at least one more question in the study material that likewise lists the vertical stretch, but not the identical horizontal shrink, as the correct answer. Is it because g is originally expressed as $g(x)=2x+3$? Does this necessitate that we think of the transformation only in the vertical axis? Something to do with $y=mx+b$ where $m=2$? Many thanks.
But for every other type of curve (in general; there are always specific cases where some transformations are equivalent or can be obtained using a combination of others) they will not have the same result. Try playing with vertical scaling and horizontal shifting of $y=2^x$ to see another version of the issue you encountered.
So, why treat it as vertical scaling only? To some extent, they're really the same thing. Going up twice as fast as the same as going along at half the speed. When working with straight lines, the idea of relative rate of change is often what we are most concerned with, the vertical change per unit horizontal change. Horizontal scaling would mess with the "per unit" aspect. Vertical scaling corresponds directly to changing the rate. So, vertical scaling is a better choice on this case.
Three minutes is three units to the right on the horizontal axis, Time. If you lightly draw a vertical line up from 3, the point where the vertical line intersects the graph is the answer. Put a dot at this point and read the temperature on the vertical axis. The temperature of the gas at three minutes is approximately 35 degrees Celsius.
The graph below highlights the 5 definitions:The x axis is the horizontal axis, and x is the independent variable.
The scale of the x axis is 5.
The y axis is the vertical axis, and y is the dependent variable.
The scale of the y axis is 25.
The ordered pair is indicated by (x, y).
The x intercepts are approximately (-2.5, 0) and (27.5, 0).
The y intercept is approximately (0, 75).
This section presents an additional way to graph a line. To graph a line, you need a minimum of two points. Two special points can be used. They are the intercepts of each axis. Often the intercepts have special meanings in a mathematical model. Also covered in this section are horizontal and vertical lines.
You should write the equations for horizontal and vertical lines on a note card. You will need to graph horizontal lines in the Section 2.9 Applications of Graphs. You should review this card at least twice a week.
Explanation: 34.55 - 28.15, represents the change in cost. Cost is the dependent variable and the vertical axis.30-10, represents the change in miles. Miles are the independent variable and the horizontal axis.
The basic ideas in this section:The formula for the slope of a line is .
If the slope of a line is positive, then the line is increasing or rising from left to right.
If the slope of a line is negative, then the line is decreasing or falling from left to right.
The slope of a horizontal line is zero.
The slope of a vertical line is undefined.
In the slope-intercept equation, y = mx + b, m is the slope, and (0,b) is the y intercept.
The intersection is where two lines cross. The intersection is found by setting the two equations equal to each other and solving algebraically for x (or the independent variable). Then the y coordinate (or dependent variable) is found by substituting the solution into one of the equations.
The important features of a graph are:Vertex: The high or low point.
Intercept: The point at which the graph crosses the horizontal or vertical axis.
Intersection: The point where two graphs meet.
Slope: The steepness of a line and the average rate of change.
Study Tips:Make sure you have done all of the homework exercises.
Practice the review test starting on the next page by placing yourself under realistic exam conditions.
Find a quiet place and use a timer to simulate the class length.
Write your answers in your homework notebook or make a copy of the test. You may then re-take the exam for extra practice.
Check your answers. The answers to the review test are on page 451.
There is an additional exam available on the Beginning Algebra web page, see page xi.
Review often. Do NOT wait until the night before to study.
Next, notice that lines A and B slant up as you move from left to right. We say these two lines have a positive slope. Line C slants down from left to right. Line C has a negative slope. Using two of the points on the line, you can find the slope of the line by finding the rise and the run. The vertical change between two points is called the rise, and the horizontal change is called the run. The slope equals the rise divided by the run: [latex] \displaystyle \textSlope =\frac\textrise\textrun[/latex].
Two non-vertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other. If the slope of the first equation is 4, then the slope of the second equation will need to be [latex]-\frac14[/latex] for the lines to be perpendicular.
In the case where one of the lines is vertical, the slope of that line is undefined and it is not possible to calculate the product with an undefined number. When one line is vertical, the line perpendicular to it will be horizontal, having a slope of zero ([latex]m=0[/latex]).
Geometry taught us that exactly one line crosses through any two points. We can use this fact in algebra as well. When drawing the graph of a line, we only need two points, and then use a straight edge to connect them. Remember, though, that lines are infinitely long: they do not start and stop at the two points we used to draw them.
Lines can be expressed algebraically as an equation that relates the $y$-values to the $x$-values. We can use the same fact that we used earlier that two points are contained in exactly one line. With only two points, we can determine the equation of a line. Before we do this, let's discuss some very important characteristics of lines: slope, $y$-intercept, and $x$-intercept.
Think of the slope of a line as its "steepness": how quickly it rises or falls from left to right. This value is indicated in the graph above as $\frac\Delta y\Delta x$, which specifies how much the line rises or falls (change in $y$) as we move from left to right (change in $x$). It is important to relate slope or steepness to the rate of vertical change per horizontal change. A popular example is that of speed, which measures the change in distance per change in time. Where a line can represent the distance traveled at various points in time, the slope of the line represents the speed. A steep line represents high speed, whereas very little steepness represents a much slower rate of travel, or low speed. This is illustrated in the graph below.
The vertical axis represents distance, and the horizontal axis represents time. The red line is steeper than the blue and green lines. Notice the distance traveled after one hour on the red line is about 5 miles. It is much greater than the distance traveled on the blue or green lines after one hour - about $1$ mile and $\frac15$, respectively. The steeper the line, the greater the distance traveled per unit of time. In other words, steepness or slope represents speed. The red lines is the fastest, with the greatest slope, and the green line is the slowest, with the smallest slope.
Slope can be classified in four ways: positive, negative, zero, and undefined slope. Positive slope means that as we move from left to right on the graph, the line rises. Negative slope means that as we move from left to right on the graph, the line falls. Zero slope means that the line is horizontal: it neither rises nor falls as we move from left to right. Vertical lines are said to have "undefined slope," as their slope appears to be some infinitely large, undefined value. See the graphs below that show each of the four slope types. 2ff7e9595c
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