Question

3.3.3 quiz: point - slope equation of a line
find a point - slope equation of the line.
a. $y - 9=-4(x + 4)$
b. $y - 9 = 4(x + 4)$
c. $y + 9=-4(x - 4)$
d. $y + 9 = 4(x - 4)$

Explanation:

Step1: Recall Point - Slope Formula

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope of the line.

Step2: Identify the Point on the Line

From the graph, we know that the point $(4,-9)$ lies on the line. So, $x_1 = 4$ and $y_1=-9$. Substituting these values into the point - slope formula, we get $y-(-9)=m(x - 4)$, which simplifies to $y + 9=m(x - 4)$.

Step3: Calculate the Slope

We can also see that the line passes through the y - intercept. From the graph, the y - intercept is at $(0,9)$? Wait, no, looking at the graph, when $x = 0$, the y - value is $9$? Wait, no, the line goes through $(0,9)$? Wait, no, the point $(4,-9)$ and let's find another point. Let's take the y - intercept. Let's assume the y - intercept is $(0,9)$? Wait, no, when $x = 0$, from the graph, the y - coordinate is $9$? Wait, no, the line is going from $(0,9)$ to $(4,-9)$. Let's calculate the slope between $(0,9)$ and $(4,-9)$. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(0,9)$ and $(x_2,y_2)=(4,-9)$. Then $m=\frac{-9 - 9}{4-0}=\frac{-18}{4}=- 4.5$? Wait, no, maybe I misread the y - intercept. Wait, the graph has a point at $(4,-9)$ and the line intersects the y - axis at $(0,9)$? Wait, no, the scale: the vertical axis, the distance between the marks. Wait, maybe the y - intercept is $(0,9)$ and the point $(4,-9)$. Let's recalculate the slope. $m=\frac{-9 - 9}{4-0}=\frac{-18}{4}=-4.5$? No, that can't be. Wait, maybe the y - intercept is $(0,9)$? Wait, no, the options have slope - 4 or 4. Let's check the options. Our equation is $y + 9=m(x - 4)$. So we need to find $m$. Let's take two points: the y - intercept (0, b) and (4,-9). Let's assume the y - intercept is (0,9). Then the slope $m=\frac{-9 - 9}{4-0}=\frac{-18}{4}=- 4.5$, but the options have slope - 4 or 4. Wait, maybe the y - intercept is (0,7)? No, maybe I made a mistake. Wait, the line passes through (0,9) and (4,-9). Wait, no, the difference in y is $-9-9=-18$, difference in x is $4 - 0 = 4$, so slope is $-18/4=-4.5$, but the options have slope - 4. Wait, maybe the y - intercept is (0,7)? No, let's look at the options. The options have $m=-4$ or $m = 4$. Let's check the options. Our equation is $y + 9=m(x - 4)$. So we can eliminate options A and B because they have $y-9$ instead of $y + 9$. Now, between C and D. Option C has $m=-4$ and option D has $m = 4$. Let's calculate the slope between (0,9) and (4,-9). Wait, if $m=-4$, then using the point - slope form with $(4,-9)$, $y+9=-4(x - 4)$. Let's check if this line passes through (0,9). Substitute $x = 0$ into $y+9=-4(0 - 4)=16$, then $y=16 - 9 = 7$. No, that's not 9. Wait, maybe the y - intercept is (0,7). Then if $x = 0$, $y=7$. Then $y+9=-4(x - 4)$. Substitute $x = 0$: $y+9=-4(-4)=16$, $y=7$. Yes! So when $x = 0$, $y = 7$. Wait, maybe the y - intercept is (0,7). Then the slope between (0,7) and (4,-9) is $m=\frac{-9 - 7}{4-0}=\frac{-16}{4}=-4$. Ah, that's correct. So the slope $m=-4$. So the equation is $y + 9=-4(x - 4)$.

Answer:

C. $y + 9=-4(x - 4)$