Algebra II - Graphing Linear Functions

Determine where the graphs of the following equations will intersect.

Explanation

We can solve the system of equations using the substitution method.

Solve for in the second equation.

Substitute this value of into the first equation.

Now we can solve for .

$y=\frac{14}{-7}=-2$

Solve for using the first equation with this new value of .

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

The solution is the ordered pair .

What is the equation of the line passing through (-1,4) and (2,6)?

$y=\frac{2}{3}x+\frac{14}{3}$

$y=-\frac{2}{3}x+\frac{14}{3}$

$y=\frac{2}{3}x-\frac{14}{3}$

$y=3x+\frac{14}{3}$

$y=\2x+\frac{14}{3}$

Explanation

To find the equation of this line, first find the slope. Recall that slope is the change in y over the change in x: $\frac{6-4}{2+1}=\frac{2}{3}$ . Then, pick a point and use the slope to plug into the point-slope formula ( $y-y_{1}=m(x-x_{1})$ ): $y-6=\frac{2}{3}(x-2)$ . Distribute and simplify so that you solve for y: $y=\frac{2}{3}x+\frac{14}{3}$ .

Find the slope-intercept form of an equation of the line that has a slope of $\frac{-3}{4}$ and passes through .

$y=\frac{-3}{4}x+8$

$y=\frac{3}{4}x+8$

$y=\frac{-3}{4}x-8$

$y=\frac{4}{3}x+8$

Explanation

Since we know the slope and we know a point on the line we can use those two piece of information to find the y-intercept.

$2=\frac{-3}{4}(8)+b$

$y=\frac{-3}{4}x+8$

Screen_shot_2014-12-24_at_2.55.25_pm

What is the equation of the above line?

Explanation

The equation of a line is with m being the slope and b being the y intercept. The y-intercept is at , so . The x-intercept is , so after plugging in the equation becomes , simplifying to .

Inequality

Which of the following inequalities is graphed above?

$y \leq - \frac{3}{8} x + 3$

$y \geq - \frac{3}{8} x + 3$

$y \leq - \frac{3}{8} x +8$

$y \geq - \frac{8}{3} x - 3$

$y \geq - \frac{8}{3} x - 8$

Explanation

First, we determine the equation of the boundary line. This line includes points and , so the slope can be calculated as follows:

$m = \frac{y_{2}- y_{1}}{x_{2}- x_{1}} = \frac{3-0}{0- 8} = - \frac{3}{8}$

Since we also know the -intercept is , we can substitute $m = - \frac{3}{8}, b = 3$ in the slope-intercept form to obtain the equation of the boundary line:

$y = - \frac{3}{8} x + 3$

The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either $\leq$ or $\geq$ . To find out which one, we can test a point in the solution set - for ease, we will choose :

_____ $- \frac{3}{8} x + 3$

_____ $- \frac{3}{8}\cdot 0 + 3$

_____

0 is less than 3 so the correct symbol is $\leq$ .

The inequality is $y \leq - \frac{3}{8} x + 3$ .

Determine the slope of a line that has points and .

$-\frac{2}{3}$

$-\frac{3}{2}$

$\frac{2}{3}$

$\frac{3}{2}$

Explanation

Slope is the change of a line. To find this line one can remember it as rise over run. This rise over run is really the change in the y direction over the change in the x direction.

Therefore the formula for slope is as follows.

$Slope = \frac{\Delta Y}{\Delta X}=\frac{y_2-y_1}{x_2-x_1}$

Plugging in our given points

and ,

$Slope=\frac{y2-y1}{x2-x1}=\frac{5-3}{-1-2}=\frac{2}{-3}=-\frac{2}{3}$ .

What is the slope of ?

Explanation

To solve this, first put the linear equation into slope-intercept form:

Recall that in slope intercept form

the m term is the slope value.

Therefore, the slope is 2.

Where does cross the axis?

Explanation

To find where this equation crosses the axis or its -intercept, change the equation into slope intercept form.

Subtract to isolate :

Divide both sides by to completely isolate :

This form is the slope intercept form where is the slope of the line and is the -intercept.

Which of the following is the function graphed below?

Graph 20150731 142249

$\small y=2x-5$

$\small y=2x+5$

$\small y=\frac{1}{2}x-5$

$\small y=\frac{1}{2}x+5$

$\small y=-2x-5$

Explanation

This function is linear (a line), so we must remember that we can represent lines algebraically using y=mx+b, where m is the slope and b is the y-intercept.

Looking at the graph, we can tell immediately that the y-intercept is -5, because the line crosses(intercepts) the y-axis at -5.

To find the slope, we need two points, $\small \small (x_{1},y_{1}),(x_{2},y_{2})$ and the following formula:

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

For the sake of the example, choose (0,-5) and (2,-1). We can see that the graph clearly passes through each of these points. Any two points will do, however. Substituting each of the values into the slope formula yields m=2.

Thus, our final answer is $\small \small y=2x-5$

How many -intercepts does the graph of the function

$f(x) = - x^{2} + 7x +11$

have?

Two

One

Zero

Four

Cannot be determined

Explanation

The graph of a quadratic function $f(x) = ax^{2}+ bx + c$ has an -intercept at any point at which , so, first, set the quadratic expression equal to 0:

$- x^{2} + 7x +11 = 0$

The number of -intercepts of the graph is equal to the number of real zeroes of the above equation, which can be determined by evaluating the discriminant of the equation $b^{2} -4ac$ . Set , and evaluate: