Algebra 1 : Functions and Lines

Example Questions

Example Question #1 : How To Graph A Function

Which graph depicts a function?

Explanation:

A function may only have one y-value for each x-value.

The vertical line test can be used to identify the function. If at any point on the graph, a straight vertical line intersects the curve at more than one point, the curve is not a function.

Example Question #1 : Functions As Graphs

The graph below is the graph of a piece-wise function in some interval.  Identify, in interval notation, the decreasing interval.

Explanation:

As is clear from the graph, in the interval between  ( included) to , the  is constant at  and then from ( not included) to  ( not included), the  is a decreasing function.

Example Question #1 : Graphing Polynomial Functions

Which equation best represents the following graph?

None of these

Explanation:

We have the following answer choices.

The first equation is a cubic function, which produces a function similar to the graph. The second equation is quadratic and thus, a parabola. The graph does not look like a prabola, so the 2nd equation will be incorrect. The third equation describes a line, but the graph is not linear; the third equation is incorrect. The fourth equation is incorrect because it is an exponential, and the graph is not an exponential. So that leaves the first equation as the best possible choice.

Example Question #1 : Asymptotes

What is the horizontal asymptote of the graph of the equation  ?

Explanation:

The asymptote of this equation can be found by observing that  regardless of . We are thus solving for the value of as approaches zero.

So the value that  cannot exceed is , and the line  is the asymptote.

Example Question #1 : How To Graph An Exponential Function

What is/are the asymptote(s) of the graph of the function

?

Explanation:

An exponential equation of the form  has only one asymptote - a horizontal one at . In the given function, , so its one and only asymptote is .

Explanation:

Example Question #1 : How To Graph A Two Step Inequality

Which graph depicts the following inequality?

No real solution.

Explanation:

Let's put the inequality in slope-intercept form to make it easier to graph:

The inequality is now in slope-intercept form. Graph a line with slope  and y-intercept .

Because the inequality sign is greater than or equal to, a solid line should be used.

Next, test a point. The origin  is good choice. Determine if the following statement is true:

The statement is false. Therefore, the section of the graph that does not contain the origin should be shaded.

Example Question #31 : Functions And Lines

What is the minimum possible value of the expression below?

The expression has no minimum value.

Explanation:

We can determine the lowest possible value of the expression by finding the -coordinate of the vertex of the parabola graphed from the equation . This is done by rewriting the equation in vertex form.

The vertex of the parabola  is the point .

The parabola is concave upward (its quadratic coefficient is positive), so  represents the minimum value of . This is our answer.

Example Question #1 : Graphing Parabolas

What is the vertex of the function ? Is it a maximum or minimum?

; minimum

; minimum

; maximum

; maximum

; minimum

Explanation:

The equation of a parabola can be written in vertex form: .

The point  in this format is the vertex. If  is a postive number the vertex is a minimum, and if  is a negative number the vertex is a maximum.

In this example, . The positive value means the vertex is a minimum.

Example Question #1 : Understand Linear And Nonlinear Functions: Ccss.Math.Content.8.F.A.3

Which of the graphs best represents the following function?