Compare the function 3x+2y=12 to the function graphed in the following image.
Which of the following is a true statement?
A. the y-intercept of the graphed function is greater than the y- intercept of the function 3x+2y=12
B. the x-intercept of the graphed function is greater than the x- intercept of the function 3x+2y=12
C. the two functions have the same x-intercept
D. the two functions have the same y-intercept
To determine the relationship between the given function 3x+2y=12 and the graph, let's analyze the options:
A. the y-intercept of the graphed function is greater than the y- intercept of the function 3x+2y=12
Since the y-intercept of the graphed function is the value of y when x=0, we can see from the image that the graphed function's y-intercept is 2. On the other hand, in the equation 3x+2y=12, when x=0, we have 2y=12, which means y=6. Therefore, the y-intercept of the function 3x+2y=12 is greater than the y-intercept of the graphed function. As a result, option A is false.
B. the x-intercept of the graphed function is greater than the x- intercept of the function 3x+2y=12
Similar to the previous analysis, the x-intercept of the graphed function is the value of x when y=0. From the image, we can observe that the graphed function's x-intercept is approximately 5. On the other hand, in the equation 3x+2y=12, when y=0, we can solve for x: 3x+2(0)=12. This simplifies to 3x=12 and x=4. Therefore, the x-intercept of the function 3x+2y=12 is less than the x-intercept of the graphed function, making option B false.
C. the two functions have the same x-intercept
We just established in the analysis of option B that the two functions have different x-intercepts. Therefore, option C is false.
D. the two functions have the same y-intercept
We established earlier that the y-intercept of the graphed function is 2. When we plug in x=0 in the equation 3x+2y=12, we get 2y=12, which simplifies to y=6. Therefore, the two functions have different y-intercepts. As a result, option D is false.
None of the statements are true, so the correct answer is none of the above.