Graphing Y+4=4(x+1): Key Points Explained

Jan 8, 2026 by Andrew McMorgan 42 views

Hey guys! Today, we're diving deep into the world of linear equations and graphing. We've got a specific equation, $y+4=4(x+1)$ , and our mission is to figure out which statements about its graph are true. This isn't just about getting the right answer; it's about understanding why that answer is right. We'll break down the equation, explore its properties, and see how to identify key points on the graph. So, grab your notebooks, and let's get started on unraveling this mathematical puzzle. Understanding these concepts is crucial for acing your math classes and building a solid foundation for more advanced topics.

Understanding the Equation's Form

First off, let's take a good look at the equation: $y+4=4(x+1)$ . This form might look a little unfamiliar if you're used to the standard $y=mx+b$ (slope-intercept form) or $Ax+By=C$ (standard form). However, this equation is actually in point-slope form, which is $y-y_1 = m(x-x_1)$ . Here, $m$ represents the slope of the line, and $(x_1, y_1)$ is a specific point that the line passes through. Recognizing this form is the first big step to understanding the graph. In our equation, $y+4=4(x+1)$ , we can see that $m=4$ . Now, for the point $(x_1, y_1)$ , we need to be a little careful. The general form is $y-y_1$ , but we have $y+4$ . This means $y - (-4)$ , so $y_1 = -4$ . Similarly, the general form is $x-x_1$ , and we have $x+1$ , which means $x - (-1)$ , so $x_1 = -1$ . Therefore, the point-slope form directly tells us that the line has a slope of 4 and passes through the point (-1, -4). This is a super valuable piece of information that we can use to check our options later. It's all about recognizing patterns and transforming the equation into a form that reveals its properties easily. Don't be intimidated by different forms; they're just different ways of saying the same thing about the line!

Converting to Slope-Intercept Form

While the point-slope form gives us immediate insight, converting the equation to the slope-intercept form ( $y=mx+b$ ) can also be very helpful. This form directly gives us the slope ( $m$ ) and the y-intercept ( $b$ ). Let's do that for $y+4=4(x+1)$ . Our goal is to isolate $y$ on one side of the equation.

Distribute the 4 on the right side: $y+4 = 4x + 4$
Subtract 4 from both sides to isolate y: $y + 4 - 4 = 4x + 4 - 4$ $y = 4x$

Wow, look at that! The equation simplifies beautifully to $y=4x$ . This tells us a few more critical things about our line:

The slope ( $m$ ) is 4. This matches what we found from the point-slope form, which is a good sign!
The y-intercept ( $b$ ) is 0. Remember, the slope-intercept form is $y=mx+b$ . In $y=4x$ , the value of $b$ is implicitly 0. This means the line passes through the origin (0, 0). This is a really important characteristic of the graph.

So, by converting to slope-intercept form, we've confirmed the slope and discovered the y-intercept. This gives us two solid points of reference: the point (-1, -4) from the point-slope form, and the point (0, 0) from the slope-intercept form. Now we're well-equipped to evaluate the given options.

Evaluating the Options

Let's examine each statement provided and see if it aligns with our findings. We know our line has a slope of 4 and passes through the points (-1, -4) and (0, 0).

Option A: The graph is a line that goes through the points (1, -4) and (0, 0).

Check point (0, 0): We found that our line does pass through (0, 0) because the y-intercept is 0. This part is correct.
Check point (1, -4): Does the line pass through (1, -4)? Let's plug these values into our simplified equation $y=4x$ . If $x=1$ , then $y = 4(1) = 4$ . So, the point on the line when $x=1$ is actually (1, 4), not (1, -4). Therefore, this statement is false.

Option B: The graph is a line that goes through the points (1, 4) and (2, 8).

Check point (1, 4): We just figured this out! When $x=1$ , $y=4(1)=4$ . So, the line does pass through (1, 4). This part is correct.
Check point (2, 8): Let's plug $x=2$ into $y=4x$ . $y = 4(2) = 8$ . So, the line also passes through (2, 8). This part is also correct.

Since the line passes through both points mentioned in Option B, this statement is true.

Option C: The graph is a line that goes through the points (-1, -4) and (0, 0).

Check point (-1, -4): As we deduced from the point-slope form, the equation $y+4=4(x+1)$ inherently means the line passes through (-1, -4). Let's verify with $y=4x$ : If $x=-1$ , then $y=4(-1)=-4$ . So, the point (-1, -4) is indeed on the line. This part is correct.
Check point (0, 0): We also found from the slope-intercept form that the line passes through the origin (0, 0). This part is correct.

Since the line passes through both points mentioned in Option C, this statement is also true.

Final Conclusion

We've analyzed the equation $y+4=4(x+1)$ and found that it simplifies to $y=4x$ . This line has a slope of 4 and passes through the origin (0, 0). It also inherently passes through the point (-1, -4) from its point-slope form.

When evaluating the options:

Option A is false because the line does not pass through (1, -4).
Option B is true because the line passes through both (1, 4) and (2, 8).
Option C is true because the line passes through both (-1, -4) and (0, 0).

Therefore, the statements that are true about the graph of the equation $y+4=4(x+1)$ are B and C. It's awesome how manipulating the equation can reveal so much about its visual representation on a graph! Keep practicing, and you'll become a graphing pro in no time.