Question

natalie was asked to find the slope of the function using the graph below. she stated that the slope was $-\frac{2}{4}$; however, her sister, jessica, stated that the slope was $-\frac{1}{4}$. who is correct?

Explanation:

Step1: Identify two points

From the graph, we can identify two clear points: the y - intercept \((0, 4)\) (wait, looking at the graph, when \(x = 0\), \(y=4\)? Wait, no, looking at the grid, when \(x = 0\), \(y = 4\)? Wait, the graph crosses the y - axis at \((0,4)\) and another point, let's take \((4, 3)\)? Wait, no, let's take two points with integer coordinates. Let's take \((0,4)\) and \((4, 3)\)? Wait, no, let's look at the grid. Let's take the y - intercept \((0,4)\) and the point where \(x = 4\), \(y=3\)? Wait, no, maybe \((0,4)\) and \((4, 3)\) is not correct. Wait, let's take two points: for example, \((0,4)\) and \((4, 3)\) gives a slope of \(\frac{3 - 4}{4-0}=\frac{- 1}{4}\). Wait, no, maybe I made a mistake. Wait, the graph is a line. Let's take two points: \((0,4)\) and \((4, 3)\) is not right. Wait, let's take \((0,4)\) and \((8, 2)\). Then the slope is \(\frac{2 - 4}{8-0}=\frac{-2}{8}=-\frac{1}{4}\). Wait, no, Natalie said \(-\frac{2}{4}\) (which is \(-\frac{1}{2}\)) and Jessica said \(-\frac{1}{4}\). Wait, let's find two points correctly. Let's take the y - intercept: when \(x = 0\), \(y = 4\) (from the graph, the line crosses the y - axis at \(y = 4\)). Then another point: when \(x = 4\), what's \(y\)? Let's see the grid. Each square is 1 unit. So from \((0,4)\), moving 4 units to the right (x increases by 4) and 1 unit down (y decreases by 1), we get \((4, 3)\). So the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3 - 4}{4-0}=\frac{-1}{4}\). Wait, but maybe another pair of points. Let's take \((0,4)\) and \((8, 2)\). Then \(m=\frac{2 - 4}{8 - 0}=\frac{-2}{8}=-\frac{1}{4}\). So Jessica is correct? Wait, no, maybe I misread the graph. Wait, the graph: let's check the y - intercept. Looking at the graph, the line crosses the y - axis at \(y = 4\)? Wait, the grid has 0 at the origin, with x from - 10 to 10 and y from - 5 to 10. Wait, maybe the y - intercept is at \((0,4)\) and when \(x = 4\), \(y = 3\), so slope is \(-\frac{1}{4}\). So Jessica is correct. Wait, but let's do it properly.

Step2: Calculate the slope

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1,y_1)\) and \((x_2,y_2)\) are two points on the line.

Let's take two points on the line: \((0,4)\) (y - intercept) and \((4, 3)\).

Then \(x_1 = 0\), \(y_1 = 4\), \(x_2=4\), \(y_2 = 3\).

\(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3 - 4}{4-0}=\frac{-1}{4}\)

Or take \((0,4)\) and \((8, 2)\):

\(m=\frac{2 - 4}{8-0}=\frac{-2}{8}=-\frac{1}{4}\)

So the slope is \(-\frac{1}{4}\), which is what Jessica stated. So Jessica is correct.

Answer:

Jessica is correct (the slope is \(-\frac{1}{4}\))