91Ó°ÊÓ

The equation given in the problem, \( y = -\frac{3}{4}x + 7 \), is written in the slope-intercept form. This form of a linear equation is particularly useful because it clearly shows two important components: the slope and the y-intercept.
The slope-intercept form is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.
The slope indicates how steep the line is and the direction it goes. In our equation, the slope is \( -\frac{3}{4} \). This means that for every 4 units we move to the right (positive direction) on the x-axis, we move 3 units down (negative direction) on the y-axis.
The y-intercept is the point where the line crosses the y-axis. In the equation \( y = -\frac{3}{4}x + 7 \), the y-intercept is 7. So, the graph of this equation crosses the y-axis at the point (0, 7).

Graphing linear equations involves plotting points and drawing the line that passes through them, based on the equation. Here’s a simple way to do it for our equation:
First, identify the y-intercept. This is where the line crosses the y-axis. For our equation \( y = -\frac{3}{4}x + 7 \), it’s at (0, 7). Plot this point on the graph.

• Next, use the slope to find another point. The slope tells us the rise over run. In our case, the slope \( -\frac{3}{4} \) means we go down 3 units and right 4 units. From (0, 7), moving right 4 units brings us to x = 4, and moving down 3 units brings us to y = 4. Plot this second point, which is (4, 4).
• Draw a line through these two points. This line is the graphical representation of the equation \( y = -\frac{3}{4}x + 7 \).

Now, you have successfully graphed the linear equation!

After graphing the linear equation, the next step is to determine if it represents a function. To do this, we use the Vertical Line Test. This test helps us check if a relation is a function by seeing if any vertical line drawn would intersect the graph at more than one point.
For the equation \( y = -\frac{3}{4}x + 7 \), if we draw vertical lines at any x-value along the graph, each line will intersect the graph at only one unique point. This confirms that our graph passes the Vertical Line Test.

• In simpler terms, a function should assign exactly one output (y-value) for each input (x-value). Since the graph of \( y = -\frac{3}{4}x + 7 \) does this, we can confidently say that it is a function.

The Vertical Line Test is a quick and effective way to check for functions, especially when dealing with linear equations like the one in this problem.