Math / Geometry

QuestionGraph the line with slope $\frac{1}{2}$ passing through the point $(-4,3)$

Studdy Solution

STEP 1

What is this asking? We need to draw a line on a graph that goes through the point $(-4,3)$ and has a slope of $\frac{1}{2}$ . Watch out! Remember slope is rise over run, and don't mix up your x and y coordinates!

STEP 2

1. Use the point-slope form
2. Convert to slope-intercept form
3. Identify the y-intercept
4. Plot the given point and the y-intercept
5. Draw the line

STEP 3

The point-slope form of a linear equation is given by $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is a given point on the line.
This form is super useful because it lets us build the equation of a line if we know its slope and a point it passes through.

STEP 4

Our slope is $m = \frac{1}{2}$ and our given point is $(-4, 3)$ , so $x_1 = -4$ and $y_1 = 3$ .
Let's plug these values into the point-slope form: $y - 3 = \frac{1}{2}(x - (-4))$ .

STEP 5

We can simplify this to $y - 3 = \frac{1}{2}(x + 4)$ .
See how subtracting a negative number turns into addition?
Sneaky, right?

STEP 6

Let's distribute the $\frac{1}{2}$ to both terms inside the parentheses: $y - 3 = \frac{1}{2} \cdot x + \frac{1}{2} \cdot 4$ , which simplifies to $y - 3 = \frac{1}{2}x + 2$ .

STEP 7

To get the slope-intercept form, we need to isolate $y$ .
We can do this by adding $3$ to both sides of the equation: $y - 3 + 3 = \frac{1}{2}x + 2 + 3$ .
This simplifies to $y = \frac{1}{2}x + 5$ .

STEP 8

Now our equation is in slope-intercept form, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

STEP 9

In our equation, $y = \frac{1}{2}x + 5$ , the y-intercept is $5$ , which means the line crosses the y-axis at the point $(0, 5)$ .

STEP 10

First, plot the given point $(-4, 3)$ .
Remember, the first number in the coordinate pair is the x-value, and the second is the y-value.

STEP 11

Next, plot the y-intercept, which is at $(0, 5)$ .

STEP 12

Now, draw a straight line that passes through both points, $(-4, 3)$ and $(0, 5)$ .
Extend the line beyond the points in both directions to show that it continues infinitely.

STEP 13

The graph of the line with slope $\frac{1}{2}$ passing through the point $(-4,3)$ is the line represented by the equation $y = \frac{1}{2}x + 5$ , plotted through the points $(-4, 3)$ and $(0, 5)$ and extending infinitely in both directions.

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