Consider the graph of the linear function below.
Which statements are true? Select ALL that apply.
(Select all that apply.)
A. The point (−4, 0) is on the line.
B. The point (0, −2) is on the line.
C. The point (1, −6) is on the line.
D. The graph represents the equation 2𝑥 − 𝑦 = 4.
E. The graph represents the equation 2𝑥 + 𝑦 = −4.
To determine which statements are true, we need to compare the given points to the equation of the linear function.
The equation of a linear function can be written in the form 𝑦 = 𝑚𝑥 + 𝑏, where 𝑚 is the slope and 𝑏 is the y-intercept.
Looking at the graph, we can see that the line passes through the point (−4, 0), which means the y-intercept is 0. This eliminates options D and E.
To find the slope of the line, we can choose two points on the line and use the formula:
slope (𝑚) = (change in y)/(change in x) = (𝑦2 − 𝑦1)/(𝑥2 − 𝑥1).
Let's choose the points (0, -2) and (1, -6). Plugging in the values, we get:
slope (𝑚) = (-6 - (-2))/(1 - 0) = -4/1 = -4.
Now, we can determine the equation of the linear function since we know the slope and y-intercept.
𝑦 = 𝑚𝑥 + 𝑏
𝑦 = -4𝑥 + 0
This simplifies to
𝑦 = -4𝑥.
Now, we can check if the given points are on the line:
A. The point (−4, 0) is on the line.
Plugging in x = -4 into the equation, we get: 𝑦 = -4(-4) = 16. So, the point (−4, 0) is not on the line. Therefore, statement A is false.
B. The point (0, −2) is on the line.
Plugging in x = 0 into the equation, we get: 𝑦 = -4(0) = 0. So, the point (0, −2) is not on the line. Therefore, statement B is false.
C. The point (1, −6) is on the line.
Plugging in x = 1 into the equation, we get: 𝑦 = -4(1) = -4. So, the point (1, −6) is not on the line. Therefore, statement C is false.
Thus, the correct statements are:
A. The point (−4, 0) is on the line. (False)
B. The point (0, −2) is on the line. (False)
C. The point (1, −6) is on the line. (False)
D. The graph represents the equation 2𝑥 − 𝑦 = 4. (False)
E. The graph represents the equation 2𝑥 + 𝑦 = −4. (False)
To determine which statements are true, we need to analyze the given information and graph of the linear function.
From the given options, let's evaluate each statement:
A. The point (−4, 0) is on the line.
To check if this point is on the line, we can substitute x = -4 and y = 0 into the equation and see if it holds true.
2𝑥 − 𝑦 = 4 becomes 2(-4) - 0 = 4 which simplifies to -8 ≠ 4.
Therefore, statement A is NOT true.
B. The point (0, −2) is on the line.
Similarly, substituting x = 0 and y = -2 into the equation, we get:
2𝑥 − 𝑦 = 4 becomes 2(0) - (-2) = 4 which simplifies to 2 + 2 = 4.
Therefore, statement B is true.
C. The point (1, −6) is on the line.
Applying the same substitution process to the equation, we have:
2𝑥 − 𝑦 = 4 becomes 2(1) - (-6) = 4 which simplifies to 2 + 6 = 4.
Therefore, statement C is true.
D. The graph represents the equation 2𝑥 − 𝑦 = 4.
By analyzing the equation, we can rewrite it in slope-intercept form (𝑦 = 𝑚𝑥 + 𝑏):
2𝑥 − 𝑦 = 4
-𝑦 = -2𝑥 + 4
𝑦 = 2𝑥 - 4
Comparing this to the general form of the linear function (𝑦 = 𝑚𝑥 + 𝑏), we can see that 𝑚 = 2 and 𝑏 = -4.
However, the equation represented by the graph is not 2𝑥 − 𝑦 = 4.
Therefore, statement D is NOT true.
E. The graph represents the equation 2𝑥 + 𝑦 = −4.
By following the same process, we rewrite the equation in slope-intercept form:
2𝑥 + 𝑦 = -4
𝑦 = -2𝑥 - 4
Comparing this to the general form of the linear function (𝑦 = 𝑚𝑥 + 𝑏), we see that 𝑚 = -2 and 𝑏 = -4.
Therefore, the equation represented by the graph is indeed 2𝑥 + 𝑦 = −4.
Thus, statement E is true.
In summary, the true statements are:
B. The point (0, −2) is on the line.
C. The point (1, −6) is on the line.
E. The graph represents the equation 2𝑥 + 𝑦 = −4.
To determine which statements are true, we need to analyze the given linear function and its graph.
Let's analyze the options one by one:
A. The point (-4, 0) is on the line.
To check this, we substitute x = -4 and y = 0 into the equation of the line. If the equation is true, then the point lies on the line.
Let's substitute the values of x and y into the equation: 2(-4) - 0 = -8
The equation 2x - y = 4 does not hold true for the point (-4, 0). So, statement A is false.
B. The point (0, -2) is on the line.
Let's substitute x = 0 and y = -2 into the equation: 2(0) - (-2) = 2
The equation 2x - y = 4 does not hold true for the point (0, -2). So, statement B is false.
C. The point (1, -6) is on the line.
We substitute x = 1 and y = -6 into the equation: 2(1) - (-6) = 8
The equation 2x - y = 4 does not hold true for the point (1, -6). So, statement C is false.
D. The graph represents the equation 2x - y = 4.
From the equation, we can observe that the coefficient of x is positive while the coefficient of y is negative. This indicates that the slope of the line is positive and the line slopes downward. We can see that the graph of the line also slops downward. Therefore, statement D is true.
E. The graph represents the equation 2x + y = -4.
From the equation, we can observe that the coefficient of x is positive while the coefficient of y is also positive. This indicates that the slope of the line is positive and the line slopes upward. However, the graph of the line slopes downward. Therefore, statement E is false.
In summary, the true statements are:
D. The graph represents the equation 2x - y = 4.