Graphing Linear Equations - Extra Practice Here with Answers
Step 1) Identify the y-intercept from the equation. Plot that point on the coordinate plane first.
Step 2) Identify the slope from the equation. Use the slope to find the next points from the y-intercept you just plotted.
Step 3) Make sure your line matches the slope (slanted up for a positive slope, slanted down for a negative slope).
Step 4) Pick a couple points that you graphed on the line and plug their coordinates into the equation for the line. This will be how you check to make sure the points you graphed are actually supposed to be on the line.
Writing Linear Equations
Step 1) Find where the graph crosses the y-axis. This will be the y-intercept. The y-intercept is always the y-coordinate when the x-coordinate = 0.
Step 2) Find the slope between any two points on the graph.
Step 3) Fit the pieces together into slope-intercept form (y=mx+b).
Step 4) Pick a couple points that were graphed on the line and plug their coordinates into the equation you found for the line. This will be how you check to make sure the equation matches the line that was graphed.
The interactive at Scale City is very helpful with understanding how scale works in the real world. Here is a link to Scale City, and you can use it with triangles and rulers to prove that the two items are similar to one another with equivalent ratios when comparing their side lengths and interior angles.
The scale factor is a synonym for constant of proportionality or rate when talking about corresponding sides of figures. It is the number that multiplies something to equal the next corresponding unit.
In the image below, the arrows show the scale factor from one triangle to the other, and vice versa. The scale factor is dependent on which figure you are multiplying, as shown in the image.
Notice that the scale factors are reciprocals of one another since 1.5 = 1 1/2 as a fraction, this is the same as 3/2 which is the reciprocal of 2/3. So the scale factor changes to its reciprocal as you change which figure you are multiplying.
You should be able to explain what each vocabulary word means and identify corresponding sides in order to set up a proportion to find a missing side.
Congruent: another word for the same or equal, often used in geometry (about shapes)
Corresponding sides: sharing the same position, sides on different shapes that share the same appearance (same angle, slope, etc) that "match" with one another once they are enlarged/minimized to be the same size
Angles: angles are usually measured in degrees to show how narrow or wide the meeting of two sides on a shape is (acute, obtuse, right)
Similar figures: Two (or more) shapes that share similar properties and share the same angles but have different side measurements
Scale factor: the number that you use to multiply/divide by to solve for the missing measurement with corresponding sides on similar figures (Scale factor is the same thing as unit rate, constant of proportionality, and slope. These terms are just used in different situations, but they are the same thing showing the rate at which two related items are moving when compared with one another.)
Proportionate: when things are equal to one another and move at the same rate (Ex. Gary buys 3 gallons of gas for $6. How much does he spend for 6 gallons of gas? We know that gas costs the same per gallon so the unit rate is $2 per gallon and we can figure out how much money he will spend on a different number of gallons of gas from the unit rate.)
Sides are always capital letters and angles are lower case.
From the image above, we see that 10 and x are corresponding sides and 6 and 12 are corresponding sides. This is because the angles are the same in both triangles so the triangles are similar figures. The scale from the left triangle to the right triangle is multiplied by 2 because we see to get from 6 to 12 is to double it. Therefore x = 20.
The video explains how to find corresponding sides. Note that having the same angles shows that a triangle is similar to another so that you know they are proportionate to one another.
There are similar figures that are written as a single shape, but they are two shapes in one. You will work these problems the same as if they were two separate shapes. It may help to draw the shapes as separate in order to better see where the corresponding sides are. The shape to the right gives an example of the two shapes in one.
In class, we discussed how to find corresponding sides on shapes that have adjacent sides. Below are 2 examples of how to identify those corresponding sides.