Answer :
To determine if Lyle's graph is correct, let's first break down the function [tex]\( f(x) = -\frac{25}{3}x + 100 \)[/tex]:
1. Finding the Slope:
The equation of the function is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, the slope [tex]\( m \)[/tex] is [tex]\( -\frac{25}{3} \)[/tex].
2. Finding the Y-Intercept:
The y-intercept is found by setting [tex]\( x = 0 \)[/tex] in the function.
- For [tex]\( f(0) = -\frac{25}{3}(0) + 100 = 100 \)[/tex], the y-intercept ([tex]\( b \)[/tex]) is 100.
- This indicates that the graph crosses the y-axis at the point [tex]\( (0, 100) \)[/tex].
3. Finding the Zero (X-Intercept):
The zero of the function is the value of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. We set the function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = -\frac{25}{3}x + 100 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -\frac{25}{3}x = -100 \implies x = \frac{-100 \cdot 3}{-25} = 12 \][/tex]
Hence, the function has a zero at [tex]\( x = 12 \)[/tex], or the graph crosses the x-axis at the point [tex]\( (12, 0) \)[/tex].
Now we evaluate the provided options:
[tex]\[ \begin{align*} 1. & \text{ Yes, because the slope is } -\frac{25}{3} \text{ and the zero is at } (0,100). \\ & \quad \text{This is incorrect. The point } (0,100) \text{ is the y-intercept, not the zero.} \\ 2. & \text{ No, the zero is correct at } (12,0) \text{ but the slope should be } -\frac{5}{3}. \\ & \quad \text{This is incorrect. The slope is indeed } -\frac{25}{3} \text{.} \\ 3. & \text{ No, the slope should be } -\frac{5}{3} \text{ and the zero should be at } (0,100). \\ & \quad \text{This is incorrect. Both the slope and the zero are correctly given by the function.} \\ 4. & \text{ Yes, because the slope is } -\frac{25}{3} \text{ and the zero is at } (12,0). \\ & \quad \text{This is correct. The provided slope matches the function and the zero matches the solved intercept.} \end{align*} \][/tex]
Based on the explanation and the verification, the correct answer is:
Yes, because the slope is [tex]\( -\frac{25}{3} \)[/tex] and the zero is at [tex]\( (12,0) \)[/tex].
1. Finding the Slope:
The equation of the function is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, the slope [tex]\( m \)[/tex] is [tex]\( -\frac{25}{3} \)[/tex].
2. Finding the Y-Intercept:
The y-intercept is found by setting [tex]\( x = 0 \)[/tex] in the function.
- For [tex]\( f(0) = -\frac{25}{3}(0) + 100 = 100 \)[/tex], the y-intercept ([tex]\( b \)[/tex]) is 100.
- This indicates that the graph crosses the y-axis at the point [tex]\( (0, 100) \)[/tex].
3. Finding the Zero (X-Intercept):
The zero of the function is the value of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. We set the function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = -\frac{25}{3}x + 100 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -\frac{25}{3}x = -100 \implies x = \frac{-100 \cdot 3}{-25} = 12 \][/tex]
Hence, the function has a zero at [tex]\( x = 12 \)[/tex], or the graph crosses the x-axis at the point [tex]\( (12, 0) \)[/tex].
Now we evaluate the provided options:
[tex]\[ \begin{align*} 1. & \text{ Yes, because the slope is } -\frac{25}{3} \text{ and the zero is at } (0,100). \\ & \quad \text{This is incorrect. The point } (0,100) \text{ is the y-intercept, not the zero.} \\ 2. & \text{ No, the zero is correct at } (12,0) \text{ but the slope should be } -\frac{5}{3}. \\ & \quad \text{This is incorrect. The slope is indeed } -\frac{25}{3} \text{.} \\ 3. & \text{ No, the slope should be } -\frac{5}{3} \text{ and the zero should be at } (0,100). \\ & \quad \text{This is incorrect. Both the slope and the zero are correctly given by the function.} \\ 4. & \text{ Yes, because the slope is } -\frac{25}{3} \text{ and the zero is at } (12,0). \\ & \quad \text{This is correct. The provided slope matches the function and the zero matches the solved intercept.} \end{align*} \][/tex]
Based on the explanation and the verification, the correct answer is:
Yes, because the slope is [tex]\( -\frac{25}{3} \)[/tex] and the zero is at [tex]\( (12,0) \)[/tex].
