Question
The equation of a linear function, g, and a table of values of a linear function, h, are shown.
g(x)=12x+2
function table with values: (-1, -4), (0, -2), (1, 0), and (2, 2)
Which statement correctly compares the slopes and y-intercepts of the two functions?
Responses
Function g has a greater y-intercept and a slope that is 14
times the slope of function h.
Function g has a lesser y-intercept and a slope that is 14
times the slope of function h.
Function g has a lesser y-intercept and a slope that is 4 times the slope of function h.
Function g has a greater y-intercept and a slope that is 4 times the slope of function h.
g(x)=12x+2
function table with values: (-1, -4), (0, -2), (1, 0), and (2, 2)
Which statement correctly compares the slopes and y-intercepts of the two functions?
Responses
Function g has a greater y-intercept and a slope that is 14
times the slope of function h.
Function g has a lesser y-intercept and a slope that is 14
times the slope of function h.
Function g has a lesser y-intercept and a slope that is 4 times the slope of function h.
Function g has a greater y-intercept and a slope that is 4 times the slope of function h.
Answers
GPT-4o mini
To compare the slopes and y-intercepts of the two functions, let's start by identifying the necessary components for both functions.
**Function g:**
- Given: \( g(x) = 12x + 2 \)
- Slope (m) of g = 12
- Y-intercept (b) of g = 2
**Function h:**
- From the table of values:
- (-1, -4)
- (0, -2)
- (1, 0)
- (2, 2)
To find the slope of function h, we can use two points from the table. Let’s take the points (0, -2) and (1, 0):
\[
\text{Slope (m)} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-2)}{1 - 0} = \frac{2}{1} = 2
\]
So, the slope of function h is 2.
The y-intercept of function h can be observed directly from the table:
- When \( x = 0 \), \( y = -2 \); thus, the y-intercept of function h is -2.
**Comparison:**
- Slope of g = 12
- Slope of h = 2
- Y-intercept of g = 2
- Y-intercept of h = -2
Now we can evaluate the statements:
1. **Function g has a greater y-intercept and a slope that is 14 times the slope of function h.**
- Slope comparison: \( 12 \) (g) is not \( 14 \times 2 \) (h), since \( 14 \times 2 = 28 \). This statement is incorrect.
2. **Function g has a lesser y-intercept and a slope that is 14 times the slope of function h.**
- Both parts are incorrect: g does not have a lesser intercept, and the slope relationship is incorrect. This statement is incorrect.
3. **Function g has a lesser y-intercept and a slope that is 4 times the slope of function h.**
- Y-intercept comparison: \( 2 \) (g) is not lesser than \( -2 \) (h), and \( 12 \) is not \( 4 \times 2 \) (h) since \( 4 \times 2 = 8 \). This statement is incorrect.
4. **Function g has a greater y-intercept and a slope that is 4 times the slope of function h.**
- Y-intercept comparison: \( 2 \) (g) is greater than \( -2 \) (h) – this part is correct.
- Slope comparison: \( 12 \) is indeed \( 4 \times 2 \) (h) since \( 4 \times 2 = 8 \) is incorrect.
Since the last part of this statement is incorrect, we must examine others, but we already know \( 12 \) is not \( 4 \) times \( 2 \).
Thus, none of the statements correctly represent the relationships of slopes and y-intercepts between the functions g and h. The correct conclusion is:
- Function g has a greater y-intercept and a slope that is 6 times the slope of function h.
Please check this before concluding the options since no options provide the correct comparison of slopes according to inputs.
**Function g:**
- Given: \( g(x) = 12x + 2 \)
- Slope (m) of g = 12
- Y-intercept (b) of g = 2
**Function h:**
- From the table of values:
- (-1, -4)
- (0, -2)
- (1, 0)
- (2, 2)
To find the slope of function h, we can use two points from the table. Let’s take the points (0, -2) and (1, 0):
\[
\text{Slope (m)} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-2)}{1 - 0} = \frac{2}{1} = 2
\]
So, the slope of function h is 2.
The y-intercept of function h can be observed directly from the table:
- When \( x = 0 \), \( y = -2 \); thus, the y-intercept of function h is -2.
**Comparison:**
- Slope of g = 12
- Slope of h = 2
- Y-intercept of g = 2
- Y-intercept of h = -2
Now we can evaluate the statements:
1. **Function g has a greater y-intercept and a slope that is 14 times the slope of function h.**
- Slope comparison: \( 12 \) (g) is not \( 14 \times 2 \) (h), since \( 14 \times 2 = 28 \). This statement is incorrect.
2. **Function g has a lesser y-intercept and a slope that is 14 times the slope of function h.**
- Both parts are incorrect: g does not have a lesser intercept, and the slope relationship is incorrect. This statement is incorrect.
3. **Function g has a lesser y-intercept and a slope that is 4 times the slope of function h.**
- Y-intercept comparison: \( 2 \) (g) is not lesser than \( -2 \) (h), and \( 12 \) is not \( 4 \times 2 \) (h) since \( 4 \times 2 = 8 \). This statement is incorrect.
4. **Function g has a greater y-intercept and a slope that is 4 times the slope of function h.**
- Y-intercept comparison: \( 2 \) (g) is greater than \( -2 \) (h) – this part is correct.
- Slope comparison: \( 12 \) is indeed \( 4 \times 2 \) (h) since \( 4 \times 2 = 8 \) is incorrect.
Since the last part of this statement is incorrect, we must examine others, but we already know \( 12 \) is not \( 4 \) times \( 2 \).
Thus, none of the statements correctly represent the relationships of slopes and y-intercepts between the functions g and h. The correct conclusion is:
- Function g has a greater y-intercept and a slope that is 6 times the slope of function h.
Please check this before concluding the options since no options provide the correct comparison of slopes according to inputs.