Question

Use the image to answer the question.

A coordinate plane's x-axis ranges from negative 4 to 4 and its y-axis ranges from negative 6 to 8, both in unit increments. A line is plotted sloping down and to the right. It intercepts the y-axis at left parenthesis 0 comma 2 right parenthesis.

Which option gives the point-slope form of the equation of the graphed line?

(1 point)
Responses

y=12x−2.5
y equals Start Fraction 1 over 2 End Fraction x minus 2.5

y+4=12(x−3)
y plus 4 equals Start Fraction 1 over 2 End Fraction left parenthesis x minus 3 right parenthesis

y+2=12(x−1)
y plus 2 equals Start Fraction 1 over 2 End Fraction left parenthesis x minus 1 right parenthesis

x−2y=5

Answer 1

To determine the point-slope form of the equation of the line based on its y-intercept at (0, 2) and its slope, we first need to identify the slope from the line's downward slope to the right. The given options mention a slope of \( \frac{1}{2} \).

The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line.

Using the y-intercept (0, 2):
- \( x_1 = 0 \)
- \( y_1 = 2 \)
- \( m = \frac{1}{2} \)

We can substitute these values into the point-slope form:
\[
y - 2 = \frac{1}{2}(x - 0)
\]
This simplifies to:
\[
y - 2 = \frac{1}{2}x
\]

Now, to match the options given:

1. \( y = \frac{1}{2}x - 2 \)
2. \( y + 4 = \frac{1}{2}(x - 3) \)
3. \( y + 2 = \frac{1}{2}(x - 1) \)
4. \( x - 2y = 5 \)

The corresponding point-slope form of the equation we derived \( y - 2 = \frac{1}{2}x \) does not directly match the options, but if we rearrange it, it leads us to a similar structure.

Next, let’s derive a point-slope form to see how we might match the options:

1. Rearranging gives us:
\[
y = \frac{1}{2}x + 2
\]

None of the equations align perfectly. However, the line meets certain conditions consistent with one of the derived options.

Thus, the answer that best aligns with the point-slope structure while allowing for shifts that maintain the slope provided is:

**y + 4 = \(\frac{1}{2}\)(x - 3)**.

This option (2) can represent movement of point-slope manipulation from a different point, but fundamentally derives from our overall linear form.

So that would be the best option according to the slope values and intersections described.