PSAT Math - How to graph a line

Psat1question

What is the equation of the line in the graph above?

Explanation

In order to find the equation of a line in slope-intercept form $\left( y = mx+b$ , where is the slope and is the y-intercept), one must know or otherwise figure out the slope of the line (its rate of change) and the point at which it intersects the y-axis. By looking at the graph, you can see that the line crosses the y-axis at . Therefore, .

Slope is the rate of change of a line, which can be calculated by figuring out the change in y divided by the change in x, using the formula

$m = \frac{\Delta y}{\Delta x} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ .

When looking at a graph, you can pick two points on a graph and substitute their x- and y-values into that equation. On this graph, it's easier to choose points like and . Plug them into the equation, and you get

$m = \frac{1+1}{1-0} = \frac{2}{1} = 2$

Plugging in those values for and in the equation, and you get

What are the x- and y- intercepts of the equation ?

$\left ( \frac{1}{2},0 \right )\textup{and}\left ( 0,-2\right )$

$\left ( -2,0\right )\ \textup{and}\ \left ( 0,\frac{1}{2}\right )$

$\left ( 4,0\right )\ \textup{and}\ \left ( 0,-2\right )$

$\left ( -2,0\right )\ \textup{and}\ \left ( 0,4\right )$

$\left ( -2,0\right )\ \textup{and}\left ( 0,-4\right )$

Explanation

Answer: (1/2,0) and (0,-2)

Finding the y-intercept: The y-intercept is the point at which the line crosses tye y-axis, meaning that x = 0 and the format of the ordered pair is (0,y) with y being the y-intercept. The equation is in slope-intercept () form, meaning that the y-intercept, b, is actually given in the equation. b = -2, which means that our y-intercept is -2. The ordered pair for expressing this is (0,-2)

Finding the x-intercept: To find the x-intercept of the equation , we must find the point where the line of the equation crosses the x-axis. In other words, we must find the point on the line where y is equal to 0, as it is when crossing the x-axis. Therefore, substitute 0 into the equation and solve for x:

The x-interecept is therefore (1/2,0).

A line graphed on the coordinate plane below. Graph_of_y_-2x_4

Give the equation of the line in slope intercept form.

$\dpi{100} \small y=-2x+4$

$\dpi{100} \small y=2x+4$

$\dpi{100} \small y=2x-4$

$\dpi{100} \small y=-2x-4$

$\dpi{100} \small y=-x+4$

Explanation

The slope of the line is $\dpi{100} \small -2$ and the y-intercept is $\dpi{100} \small 4$ .

The equation of the line is $\dpi{100} \small y=-2x+4$ .

The equation represents a line. This line does NOT pass through which of the four quadrants?

III

Cannot be determined

Explanation

Plug in for to find a point on the line:

Thus, is a point on the line.

Plug in for to find a second point on the line:

is another point on the line.

Now we know that the line passes through the points and .

A quick sketch of the two points reveals that the line passes through all but the third quadrant.

Which of the following could be the equation of the line shown in this graph?

Line

$y=-\frac{2}{3}x +1$

$y=-\frac{5}{7}x -1$

$y=-\frac{7}{6}x -4$

$y=\frac{2}{3}x +2$

$y=\frac{4}{3}x -5$

Explanation

The line in the diagram has a negative slope and a positive y-intercept. It has a negative slope because the line moves from the upper left to the lower right, and it has a positive y-intercept because the line intercepts the y-axis above zero.

The only answer choice with a negative slope and a positive y-intercept is

$y=-\frac{2}{3}x +1$

Graph_of_y_-x_3

Give the equation of the curve.

$\dpi{100} \small y=-x^{3}$

$\dpi{100} \small y=x^{3}$

None of the other answers

$\dpi{100} \small y=x^{4}$

$\dpi{100} \small y=-x^{2}$

Explanation

Graph_of_x_3 This is the parent graph of $\dpi{100} \small x^{3}$ . Since the graph in question is negative, then we flip the quadrants in which it will approach infinity. So the graph of $\dpi{100} \small y=-x^{3}$ will start in quadrant 2 and end in 4.

How to graph a line

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