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?

Short 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) withexactly 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.

Step-by-step solution

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) withexactly 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.