How to graph a line - SAT Math

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Question

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

Answer

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.

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Question

A line graphed on the coordinate plane below.

Give the equation of the line in slope intercept form.

Answer

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

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Question

Give the equation of the curve.

Answer

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.

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Question

Billy set up a ramp for his toy cars. He did this by taking a wooden plank and putting one end on top of a brick that was 3 inches high. He then put the other end on top of a box that was 9 inches high. The bricks were 18 inches apart. What is the slope of the plank?

Answer

The value of the slope (m) is rise over run, and can be calculated with the formula below:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The coordinates of the first end of the plank would be (0,3), given that this is the starting point of the plank (so x would be 0), and y would be 3 since the brick is 3 inches tall.

The coordinates of the second end of the plank would be (18,9) since the plank is 18 inches long (so x would be 18) and y would be 9 since the box was 9 inches tall at the other end.

From this information we know that we can assign the following coordinates for the equation:

$\left(x_{1},y_{1}\right) = (0,3)$ and $(x_{2}, y_{2})= (18,9)$

Below is the solution we would get from plugging this information into the equation for slope:

$m = \frac{9-3}{18-0}}$

This reduces to $\frac{6}{18} =\frac{1}{3}$

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Question

What is the slope of the line depicted by the graph?

Answer

Looking at the graph, it is seen that the line passes through the points (-8,-5) and (8,5).

The slope of a line through the points and can be found by setting

$x_{1} = -8,y_{1} = -5, x_{2} = 8, y_{2} = 5$ :

in the slope formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{5-(-5)}{8 - (-8)} = \frac{5+5}{8 + 8} = \frac{10}{16} = \frac{5}{8}$

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Question

Give the area of the triangle on the coordinate plane that is bounded by the lines of the equations , and .

Answer

It is necessary to find the coordinates of the vertices of the triangle, each of which is the intersection of two of the three lines.

The intersection of the lines of the equations and can be found by noting that, by substituting for in the latter equation, , making the point of intersection .

The intersection of the lines of the equations and can be found by substituting for in the latter equation and solving for :

$\frac{4x}{4} = \frac{5}{4}$

$x = 1 \frac{1}{4}$

This point of intersection is $\left ( 1 \frac{1}{4}, -2 \right )$ .

The intersection of the lines of the equations and can be found by substituting for in the latter equation and solving for :

$\frac{3x }{3}= \frac{7}{3}$

$x = 2\frac{1}{3}$

Since , $y = 2\frac{1}{3}$ , and this point of intersection is $\left ( 2\frac{1}{3}, 2\frac{1}{3} \right )$ .

The lines in question are graphed below, and the triangle they bound is shaded:

We can take the horizontal side as the base of the triangle; its length is the difference of the -coordinates:

$b = 1\frac{1}{4} - (-2 ) = 3 \frac{1}{4}$

The height is the vertical distance from this side to the opposite side, which is the difference of the -coordinates:

$h = 2 \frac{1}{3} - (-2) = 4 \frac{1}{3}$

The area is half their product:

$A = \frac{1}{2} bh = \frac{1}{2} \cdot 3 \frac{1}{4} \cdot 4 \frac{1}{3} =7 \frac{1}{24}$

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Question

What is the -intercept of the function that is depicted in the graph above?

Answer

This question tests one's ability to recognize algebraic characteristics of a graph. This particular question examines a linear function.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the general algebraic function for the given graph.

Since the graph is that of a straight line, the general algebraic form of the function is,

where

$\m=\textup{slope} \b=\textup{y-intercept}$

Step 2: Identify where the graph crosses the -axis.

Therefore the general form of the function looks like,

Step 3: Answer the question.

The -intercept is three.

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