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A linear function contains the following points.
\begin{tabular}{|c|c|}
\hline[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline-4 & 4 \\
\hline 0 & 5 \\
\hline
\end{tabular}

What are the slope and [tex]$y$[/tex]-intercept of this function?

A. The slope is [tex]$\frac{1}{4}$[/tex]. The [tex]$y$[/tex]-intercept is [tex]$(0, 5)$[/tex].
B. The slope is 4. The [tex]$y$[/tex]-intercept is [tex]$(5, 0)$[/tex].
C. The slope is -4. The [tex]$y$[/tex]-intercept is [tex]$(0, 5)$[/tex].
D. The slope is [tex]$-\frac{1}{4}$[/tex]. The [tex]$y$[/tex]-intercept is [tex]$(0, 5)$[/tex].


Sagot :

To determine the slope and [tex]\(y\)[/tex]-intercept of the linear function passing through the given points, we will follow these steps:

1. Identify the Coordinates:
The given points are [tex]\((-4, 4)\)[/tex] and [tex]\((0, 5)\)[/tex].

2. Calculate the Slope [tex]\(m\)[/tex]:
The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the given points into the slope formula:
[tex]\[ x_1 = -4, \quad y_1 = 4, \quad x_2 = 0, \quad y_2 = 5 \][/tex]
Thus,
[tex]\[ m = \frac{5 - 4}{0 - (-4)} = \frac{1}{4} \][/tex]

3. Determine the [tex]\(y\)[/tex]-Intercept:
The [tex]\(y\)[/tex]-intercept is the point where the line crosses the [tex]\(y\)[/tex]-axis. This occurs when [tex]\(x = 0\)[/tex]:
From the given points, we can see that when [tex]\(x = 0\)[/tex], [tex]\(y = 5\)[/tex]. Thus, the [tex]\(y\)[/tex]-intercept is [tex]\((0, 5)\)[/tex].

4. Verify the Answer:
We have calculated the slope to be [tex]\(\frac{1}{4}\)[/tex] and the [tex]\(y\)[/tex]-intercept to be [tex]\((0, 5)\)[/tex].

Therefore, the correct answer is:
A. The slope is [tex]\(\frac{1}{4}\)[/tex]. The [tex]\(y\)[/tex]-intercept is [tex]\((0,5)\)[/tex].