Question
Question
Use the table to answer the question.
Table of Values of x and y
x y
0 −0.5
−2
3
3 4
A parabola passes through the points provided in the table. The standard form equation for a parabola is y=ax2+bx+c
. Select the correct standard form equation for the parabola.
(1 point)
Responses
y=0.65x2−0.45x−0.5
y equals 0.65 x squared minus 0.45 x minus 0.5
y=0.48x2−0.79x−0.5
y equals 0.48 x squared minus 0.79 x minus 0.5
y=3.25x2−4.75x−0.5
y equals 3.25 x squared minus 4.75 x minus 0.5
y=−0.45x2+0.65x−0.5
Use the table to answer the question.
Table of Values of x and y
x y
0 −0.5
−2
3
3 4
A parabola passes through the points provided in the table. The standard form equation for a parabola is y=ax2+bx+c
. Select the correct standard form equation for the parabola.
(1 point)
Responses
y=0.65x2−0.45x−0.5
y equals 0.65 x squared minus 0.45 x minus 0.5
y=0.48x2−0.79x−0.5
y equals 0.48 x squared minus 0.79 x minus 0.5
y=3.25x2−4.75x−0.5
y equals 3.25 x squared minus 4.75 x minus 0.5
y=−0.45x2+0.65x−0.5
Answers
Answered by
GPT-4o mini
To determine the correct standard form equation of the parabola given the points in the table:
x | y
0 | -0.5
-2 | ?
3 | ?
3 | 4
We notice that we have one y-value missing for x = -2 and both values for x = 3 also contain a y-value of 4.
From the point (0, -0.5), we know that when x = 0, y = -0.5, which gives us the constant term \( c \) in the standard form equation \( y = ax^2 + bx + c \).
Thus, we have:
\[
c = -0.5.
\]
To find the values of \( a \) and \( b \), we can use the other points if we can determine missing values. We notice that when x = 3, y = 4. We can substitute this into the equation to find a relationship between \( a \) and \( b \):
\[
4 = a(3^2) + b(3) - 0.5
\]
\[
4 = 9a + 3b - 0.5
\]
Now let's rearrange the equation:
\[
9a + 3b = 4 + 0.5
\]
\[
9a + 3b = 4.5 \tag{1}
\]
Now, analyzing the other provided option (we'll substitute a value for x = -2):
If we have another point, we could find another equation needed to solve for both unknowns \( a \) and \( b \). Unfortunately, the values for y where x = -2 are missing.
To make further progress, we can also evaluate the other three choices given and see which matches with the known point on the parabola (0, -0.5) and the point (3, 4).
Let's check the given responses one more time with known values:
1. For \( y = 0.65x^2 - 0.45x - 0.5 \):
- Check y = -0.5 when x = 0:
\[ y = 0.65(0)^2 - 0.45(0) - 0.5 = -0.5 \text{ (correct)} \]
- Check y = 4 when x = 3:
\[ y = 0.65(3)^2 - 0.45(3) - 0.5 = 0.65(9) - 1.35 - 0.5 = 5.85 - 1.35 - 0.5 = 4.0 \text{ (correct)} \]
2. For \( y = 0.48x^2 - 0.79x - 0.5 \):
- Check x = 0:
\[ y = -0.5 \text{ (correct)} \]
- When x = 3:
\[ y = 0.48(3)^2 - 0.79(3) - 0.5 = 0.48(9) - 2.37 - 0.5 \text{ (not equal to 4)} \]
3. For \( y = 3.25x^2 - 4.75x - 0.5 \):
- x = 0:
\[ y = -0.5 \text{ (correct)} \]
- x = 3:
\[ y = 3.25(9) - 14.25 - 0.5 \text{ (not equal to 4)} \]
4. For \( y = -0.45x^2 + 0.65x - 0.5 \):
- x = 0:
\[ y = -0.5 \text{ (correct)} \]
- x = 3:
\[ y = -0.45(9) + (0.65)(3) - 0.5 \text{ (not equal to 4)} \]
After checking all responses, the correct answer is:
**y = 0.65x² - 0.45x - 0.5**
This function passes through the points (0, -0.5) and (3, 4).
x | y
0 | -0.5
-2 | ?
3 | ?
3 | 4
We notice that we have one y-value missing for x = -2 and both values for x = 3 also contain a y-value of 4.
From the point (0, -0.5), we know that when x = 0, y = -0.5, which gives us the constant term \( c \) in the standard form equation \( y = ax^2 + bx + c \).
Thus, we have:
\[
c = -0.5.
\]
To find the values of \( a \) and \( b \), we can use the other points if we can determine missing values. We notice that when x = 3, y = 4. We can substitute this into the equation to find a relationship between \( a \) and \( b \):
\[
4 = a(3^2) + b(3) - 0.5
\]
\[
4 = 9a + 3b - 0.5
\]
Now let's rearrange the equation:
\[
9a + 3b = 4 + 0.5
\]
\[
9a + 3b = 4.5 \tag{1}
\]
Now, analyzing the other provided option (we'll substitute a value for x = -2):
If we have another point, we could find another equation needed to solve for both unknowns \( a \) and \( b \). Unfortunately, the values for y where x = -2 are missing.
To make further progress, we can also evaluate the other three choices given and see which matches with the known point on the parabola (0, -0.5) and the point (3, 4).
Let's check the given responses one more time with known values:
1. For \( y = 0.65x^2 - 0.45x - 0.5 \):
- Check y = -0.5 when x = 0:
\[ y = 0.65(0)^2 - 0.45(0) - 0.5 = -0.5 \text{ (correct)} \]
- Check y = 4 when x = 3:
\[ y = 0.65(3)^2 - 0.45(3) - 0.5 = 0.65(9) - 1.35 - 0.5 = 5.85 - 1.35 - 0.5 = 4.0 \text{ (correct)} \]
2. For \( y = 0.48x^2 - 0.79x - 0.5 \):
- Check x = 0:
\[ y = -0.5 \text{ (correct)} \]
- When x = 3:
\[ y = 0.48(3)^2 - 0.79(3) - 0.5 = 0.48(9) - 2.37 - 0.5 \text{ (not equal to 4)} \]
3. For \( y = 3.25x^2 - 4.75x - 0.5 \):
- x = 0:
\[ y = -0.5 \text{ (correct)} \]
- x = 3:
\[ y = 3.25(9) - 14.25 - 0.5 \text{ (not equal to 4)} \]
4. For \( y = -0.45x^2 + 0.65x - 0.5 \):
- x = 0:
\[ y = -0.5 \text{ (correct)} \]
- x = 3:
\[ y = -0.45(9) + (0.65)(3) - 0.5 \text{ (not equal to 4)} \]
After checking all responses, the correct answer is:
**y = 0.65x² - 0.45x - 0.5**
This function passes through the points (0, -0.5) and (3, 4).