Q58. - Algebra 1

Question

In this problem, you will explore $x$- and $y$- intercepts of graphs of linear equations.

a. If possible, use a straightedge to draw a line on a coordinate plane with each of the following characteristics.

b. For which characteristics were you able to create a line and for which characteristics were you unable to create a line? Explain.

c. What must be true of the $x$- and $y$- intercepts of a line?

Step-by-Step Solution

Answer

a. The graph of a line with $x-$ and $y-$ intercept is:

The graph of a line with $x$-intercept and no $y-$intercept is:

The graph of a line with exactly 2 $x$-intercepts cannot be made.

 

The graph of a line with no $x$-intercept and $y$-intercept is:

The graph of a line with exactly 2 $y$-intercepts cannot be made.

 

b. The characteristics for which it is possible to create a line are:

(i) with $x$-intercept and $y$-intercept

(ii) with $x$-intercept and no $y$-intercept

(iii) with no $x$-intercept and $y$-intercept

This is because for a linear curve, at most one $x$-intercept and $y$-intercept can be drawn.

The characteristics for which it is impossible to create a line are:

(i) with exactly 2 $x$-intercepts

(ii) with exactly 2 $y$-intercepts

This is because for a non-linear curve, exactly 2 $x$-intercepts and exactly 2 $y$-intercepts can be drawn.

 

c. The condition that must be true for the $x$- intercept and $y$- intercept of a line is A line can have no or one $x$- intercept and a line can have no or one $y$-intercept. 

Part a. Step 1. Draw the graph of a line with x − and y − intercept and with x - intercept and no y -intercept.

The graph of a line with $x-$ and $y-$ intercept is:

The graph of a line with $x-$ intercept and no $y-$ intercept is:

Part a. Step 2. Draw the graph of a line with no x - intercept and y - intercept.

The graph of a line with no $x-$ intercept and $y-$ intercept is:

Part a. Step 3. Draw the graph of a line with exactly 2 x -intercepts and with exactly 2 y -intercepts.

The graph of a line with exactly 2 $x$-intercepts cannot be made.

 

The graph of a line with exactly 2 $y$-intercepts cannot be made.

Part b. Step 1. Determine the characteristics for which it is possible to create a line.

The characteristics for which it is possible to create a line are: 

(i) with $x$-intercept and $y$-intercept

(ii) with $x$-intercept and no $y$-intercept

(iii) with no $x$-intercept and $y$-intercept

This is because for a linear curve, at most one $x$-intercept and $y$-intercept can be drawn.

Part b. Step 2. Determine the characteristics for which it is impossible to create a line.

The characteristics for which it is impossible to create a line are:

(i) with exactly 2 $x$-intercepts

(ii) with exactly 2 $y$-intercepts

This is because for a non-linear curve, exactly 2 $x$-intercepts and exactly 2 $y$-intercepts can be drawn.

Part c. Step 1. Define the x - intercept and y - intercept of a line.

The $x-$ intercept of a line is the value of $x$ where the line cuts the $x-$axis.

The $y-$ intercept of a line is the value of $y$ where the line cuts the $y-$axis.

Part c. Step 2. Determine the condition that must be true for the x - intercept and y - intercept of a line.

The condition that must be true for the $x$- intercept and $y$- intercept of a line is A line can have no or one $x$- intercept and a line can have no or one $y$-intercept.