Question

Label the image correctly (some of the choices are wrong) : True False 4-3-6-4Increasing DecreasingSlope is =Slope is equal to = (make sure you simplify)True or False the line passes through the origin ?

Answer 1

To solve the exercise, you can take the two points highlighted on the graph and use the slope formula

`\begin{gathered} m=(y_2-y_1)/(x_2-x_1) \\ \text{ Where} \\ (x_1,y_1),(x_2,y_2)\text{ are two points }through\text{ which the line passes and} \\ m\text{ is the slope of the line} \end{gathered}`

So, if you take

`\begin{gathered} (x_1,y_1)=(-4,2) \\ (x_2,y_2)=(4,-4) \end{gathered}`

the slope of the line will be

`\begin{gathered} m=(y_2-y_1)/(x_2-x_1) \\ m=(-4-2)/(4-(-4)) \\ m=(-6)/(4+4) \\ m=(-6)/(8) \\ \text{ Simplifying} \\ m=(2\cdot-3)/(2\cdot4) \\ m=(-3)/(4) \end{gathered}`

Now, with the point slope equation you can write the equation of the line in its slope-intercept form

`\begin{gathered} y-y_1=m(x-x_1) \\ \text{ Where }(x_1,y_1)\text{ is one point through which the line passes} \end{gathered}`

So, if you take the point (-4,2) you have

`\begin{gathered} y-y_1=m(x-x_1) \\ y-2=(-3)/(4)(x-(-4)) \\ y-2=(-3)/(4)(x+4) \\ y-2=(-3)/(4)x-(3)/(4)\cdot4 \\ y-2=(-3)/(4)x-3 \\ \text{ Add 2 on both sides the equation} \\ y-2+2=(-3)/(4)x-3+2 \\ y=(-3)/(4)x-1 \end{gathered}`

Then, the equation of the line in its slope-intercept form is

`y=-(3)/(4)x-1`

And to find out if the line passes through the origin, plug x = 0 into the equation found

`\begin{gathered} y=-(3)/(4)x-1 \\ y=-(3)/(4)(0)-1 \\ y=0-1 \\ y=-1 \end{gathered}`

That means that the line passes through the point (0, -1) and not through the point (0,0). Therefore, the line does not pass through the origin, and the answer is False.

Finally, since the slope of the line is negative, then the line is decreasing.