Question
You are asked to graph a parabola on a coordinate plane given the equation y = - 1/8 x^2 + 3/4 x + 7/8. Fill in the missing value on the table for the coordinates of the parabola.
x-value
-1
3
__
11
y-value
0
2
0
-6
a. 7
b. -
c. 6
d. -6
x-value
-1
3
__
11
y-value
0
2
0
-6
a. 7
b. -
c. 6
d. -6
Answers
GPT-4o mini
To find the missing y-value for the corresponding x-value in the equation \( y = -\frac{1}{8} x^2 + \frac{3}{4} x + \frac{7}{8} \), we will substitute the unknown x-value from the table and solve for y.
Given x-values:
-1, 3, __, 11
And corresponding y-values:
0, 2, ?, -6
We can start by finding the missing x-value. The missing x-value lies between the x-values of 3 and 11. Let's start with calculating the y-value for x = 5.
1. **For x = 5**:
\[
y = -\frac{1}{8}(5^2) + \frac{3}{4}(5) + \frac{7}{8}
\]
\[
= -\frac{1}{8}(25) + \frac{15}{4} + \frac{7}{8}
\]
\[
= -\frac{25}{8} + \frac{30}{8} + \frac{7}{8}
\]
\[
= \frac{-25 + 30 + 7}{8}
\]
\[
= \frac{12}{8} = \frac{3}{2} = 1.5
\]
However, that doesn't match any of the choices provided. Let's check another value between 3 and 11 that makes sense, such as x = 7.
2. **For x = 7**:
\[
y = -\frac{1}{8}(7^2) + \frac{3}{4}(7) + \frac{7}{8}
\]
\[
= -\frac{1}{8}(49) + \frac{21}{4} + \frac{7}{8}
\]
\[
= -\frac{49}{8} + \frac{42}{8} + \frac{7}{8}
\]
\[
= \frac{-49 + 42 + 7}{8}
\]
\[
= \frac{0}{8} = 0
\]
So, for x = 7, the value of y is 0, which corresponds to choice a. Given the choices available, it is likely that since x = 5 had skipped, the nearest correlation matches the existing points of interest before it.
Thus, the missing x-value is 7, corresponding to y = 0.
**However, if we check x = 11:**
3. **For x = 11**:
\[
y = -\frac{1}{8}(11^2) + \frac{3}{4}(11) + \frac{7}{8}
\]
\[
= -\frac{121}{8} + \frac{33}{4} + \frac{7}{8}
\]
\[
= -\frac{121}{8} + \frac{66}{8} + \frac{7}{8}
\]
\[
= \frac{-121 + 66 + 7}{8}
\]
\[
= \frac{-48}{8} = -6
\]
So the y-value corresponding to x = 11 is indeed -6.
To summarize:
- The missing x-value is 7 with a y-value of 0
- For x = 11, the function outputs y = -6 and that relates to option d.
Considering only the y = -6 result, the answer is option **d. -6**.
Given x-values:
-1, 3, __, 11
And corresponding y-values:
0, 2, ?, -6
We can start by finding the missing x-value. The missing x-value lies between the x-values of 3 and 11. Let's start with calculating the y-value for x = 5.
1. **For x = 5**:
\[
y = -\frac{1}{8}(5^2) + \frac{3}{4}(5) + \frac{7}{8}
\]
\[
= -\frac{1}{8}(25) + \frac{15}{4} + \frac{7}{8}
\]
\[
= -\frac{25}{8} + \frac{30}{8} + \frac{7}{8}
\]
\[
= \frac{-25 + 30 + 7}{8}
\]
\[
= \frac{12}{8} = \frac{3}{2} = 1.5
\]
However, that doesn't match any of the choices provided. Let's check another value between 3 and 11 that makes sense, such as x = 7.
2. **For x = 7**:
\[
y = -\frac{1}{8}(7^2) + \frac{3}{4}(7) + \frac{7}{8}
\]
\[
= -\frac{1}{8}(49) + \frac{21}{4} + \frac{7}{8}
\]
\[
= -\frac{49}{8} + \frac{42}{8} + \frac{7}{8}
\]
\[
= \frac{-49 + 42 + 7}{8}
\]
\[
= \frac{0}{8} = 0
\]
So, for x = 7, the value of y is 0, which corresponds to choice a. Given the choices available, it is likely that since x = 5 had skipped, the nearest correlation matches the existing points of interest before it.
Thus, the missing x-value is 7, corresponding to y = 0.
**However, if we check x = 11:**
3. **For x = 11**:
\[
y = -\frac{1}{8}(11^2) + \frac{3}{4}(11) + \frac{7}{8}
\]
\[
= -\frac{121}{8} + \frac{33}{4} + \frac{7}{8}
\]
\[
= -\frac{121}{8} + \frac{66}{8} + \frac{7}{8}
\]
\[
= \frac{-121 + 66 + 7}{8}
\]
\[
= \frac{-48}{8} = -6
\]
So the y-value corresponding to x = 11 is indeed -6.
To summarize:
- The missing x-value is 7 with a y-value of 0
- For x = 11, the function outputs y = -6 and that relates to option d.
Considering only the y = -6 result, the answer is option **d. -6**.