What this rule means is that we should be able to graph any linear equation by figuring out two points and drawing the line between them. In practice, it's a good idea to graph at least three points. If we graph three points of a linear equation and they don't all lie on the same line, we know we did something wrong.
As much as that might rattle our delicate egos, at least we can go back and fix what we fouled up. It's better than remaining blissfully ignorant, no matter what that old poet Thomas Gray might have said.
The intercepts of a linear equation are the places where the axes catch the pass thrown by the linear equation. This is our effort to make linear equations seem remotely athletic. In reality, they have about as much physical ability as Tim Tebow. Oh, snap. In non-sports-analogy terms, the intercepts are the spots at which the axes and the graph of the linear equation overlap one another. The x -intercept is the place where the graph hits the x -axis, and the y -intercept is the place where the graph hits the y -axis.
It would be awfully confusing if it were the other way around. A linear equation may have one or two intercepts. Sometimes either the x -intercept or the y -intercept doesn't exist, or so intercept atheists would have you believe. Knowing both intercepts for a linear equation is enough information to draw the graph, provided the intercepts aren't 0. If they are 0, then our graph could be drawn any which way. If the graph goes through the origin 0, 0 , then both of the intercepts are 0 and we don't have enough information to draw the graph.
We even tried calling , but they acted as if they had no idea what we were talking about. The slope of a linear equation is a number that tells how steeply the line on our graph is climbing up or down. If we pretend the line is a mountain, it's like we're talking about the slope of a mountain. If it helps you, draw a snowcap at the top. Some mountain climbers. A ski lift. Nothing too elaborate though. We move from left to right on the x -axis, the same way that we read.
If the line gets higher as we move right, then we're climbing the mountain, so the line has a positive slope. If the line gets lower as we move right, then we're descending the mountain, so the line has a negative slope.
If we stay at the same height, then the slope is zero because we're not going up and we're not going down. Pretty boring mountain, if you ask us. Now let's find some actual numbers for slopes. Thinking of the mountains, a slope is a ratio that describes how quickly our height changes as we move over to the right. Not our actual physical height, mind you. We won't be getting shorter or taller throughout the course of these examples, even if you do feel by the end of it that you've grown.
Julie is climbing a mountain. For every 10 feet Julie travels measured along the ground , she goes 20 feet higher. What is the slope of the mountain? The slope of the mountain is. For every foot Julie travels measured along the ground , she gets 2 feet higher off the ground.
She'd be even higher off the ground if she'd worn heels, but we suppose those would have been an odd choice for mountain climbing. If we move over to the right by 1 on the x -axis, we also move up by one on the y -axis:. Find the slope of the line pictured below. If we haven't heard from you in three hours, we'll send the park ranger after you. On this line, or mountain, we move up 2 for every 3 we move over. I could write a whole blog post about that comma in the first sentence, but for now I want to focus on the question of what exactly are the student expectations entailed by this standard.
By definition, a linear function is one with a constant rate of change, that is, a function where the slope between any two points on its graph is always the same. However, the following PARCC released item suggests the possible expectation that students be able to tell if a function is linear or not purely from looking at its defining equation. By the look of it it is even more not linear! This is another example of the confusion between expressions and functions. Equivalent expressions define the same function.
Look at option C. Inverse variation is the opposite of direct variation. In the case of inverse variation, the increase of one variable leads to the decrease of another. In fact, two variables are said to be inversely proportional when an operation of change is performed on one variable and the opposite happens to the other.
As an example, the time taken for a journey is inversely proportional to the speed of travel. If your car travels at a greater speed, the journey to your destination will be shorter. Knowing that the relationship between the two variables is constant, we can show that their relationship is:. We can rearrange the above equation to place the variables on opposite sides:.
Notice that this is not a linear equation. It is impossible to put it in slope-intercept form. Thus, an inverse relationship cannot be represented by a line with constant slope. Inverse variation can be illustrated with a graph in the shape of a hyperbola, pictured below. Inversely Proportional Function: An inversely proportional relationship between two variables is represented graphically by a hyperbola. The graph of a linear function is a straight line.
Linear functions can have none, one, or infinitely many zeros. Finally, if the line is vertical or has a slope, then there will be only one zero. Zeros can be observed graphically. All lines, with a value for the slope, will have one zero. Since each line has a value for the slope, each line has exactly one zero. The zero from solving the linear function above graphically must match solving the same function algebraically. This is the same zero that was found using the graphing method.
The slope-intercept form of a line summarizes the information necessary to quickly construct its graph. One of the most common representations for a line is with the slope-intercept form. This assists in finding solutions to various problems, such as graphing, comparing two lines to determine if they are parallel or perpendicular and solving a system of equations.
Simply substitute the values into the slope-intercept form to obtain:. The value of the slope dictates where to place the next point.
Using this information, graphing is easy. The point-slope equation is another way to represent a line; only the slope and a single point are needed. Use point-slope form to find the equation of a line passing through two points and verify that it is equivalent to the slope-intercept form of the equation. The point-slope equation is a way of describing the equation of a line.
Therefore, the two equations are equivalent and either one can express an equation of a line depending on what information is given in the problem or what type of equation is requested in the problem. Plug this point and the calculated slope into the point-slope equation to get:.
Be careful if one of the coordinates is a negative. Distributing the negative sign through the parentheses, the final equation is:. Again, the two forms of the equations are equivalent to each other and produce the same line.
The only difference is the form that they are written in.
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