Whats the right answer
Which polynomial has a graph that passes through the given points?
(–3, 10), (–1, 0), (0, –5), (3, 52)
(1 point)
Responses
y = x3 – 4x2 + 2x – 5
y = x 3 – 4 x 2 + 2 x – 5
y = x3 + 4x2 – 2x – 5
y = x 3 + 4 x 2 – 2 x – 5
y = x3 + 4x2 + 2x – 5
y = x 3 + 4 x 2 + 2 x – 5
y = x4 + 4x3 – 2x2 – 5x
The correct answer is:
y = x³ + 4x² + 2x - 5
To determine which polynomial has a graph that passes through the given points, we can substitute the x and y values of each point into the equation options and see which one satisfies all the points.
Let's start with option 1:
y = x^3 - 4x^2 + 2x - 5
Substituting the x and y values of the given points:
For (–3, 10): 10 = (-3)^3 - 4(-3)^2 + 2(-3) - 5
This equation is not satisfied.
For (–1, 0): 0 = (-1)^3 - 4(-1)^2 + 2(-1) - 5
This equation is satisfied.
For (0, –5): -5 = (0)^3 - 4(0)^2 + 2(0) - 5
This equation is satisfied.
For (3, 52): 52 = (3)^3 - 4(3)^2 + 2(3) - 5
This equation is satisfied.
Option 1 satisfies 3 out of 4 points.
Let's continue with option 2:
y = x^3 + 4x^2 - 2x - 5
Substituting the x and y values of the given points:
For (–3, 10): 10 = (-3)^3 + 4(-3)^2 - 2(-3) - 5
This equation is not satisfied.
For (–1, 0): 0 = (-1)^3 + 4(-1)^2 - 2(-1) - 5
This equation is not satisfied.
For (0, –5): -5 = (0)^3 + 4(0)^2 - 2(0) - 5
This equation is not satisfied.
For (3, 52): 52 = (3)^3 + 4(3)^2 - 2(3) - 5
This equation is satisfied.
Option 2 satisfies 1 out of 4 points.
Let's continue with option 3:
y = x^3 + 4x^2 + 2x - 5
Substituting the x and y values of the given points:
For (–3, 10): 10 = (-3)^3 + 4(-3)^2 + 2(-3) - 5
This equation is not satisfied.
For (–1, 0): 0 = (-1)^3 + 4(-1)^2 + 2(-1) - 5
This equation is not satisfied.
For (0, –5): -5 = (0)^3 + 4(0)^2 + 2(0) - 5
This equation is satisfied.
For (3, 52): 52 = (3)^3 + 4(3)^2 + 2(3) - 5
This equation is not satisfied.
Option 3 satisfies 1 out of 4 points.
Lastly, let's check option 4:
y = x^4 + 4x^3 - 2x^2 - 5x
Substituting the x and y values of the given points:
For (–3, 10): 10 = (-3)^4 + 4(-3)^3 - 2(-3)^2 - 5(-3)
This equation is not satisfied.
For (–1, 0): 0 = (-1)^4 + 4(-1)^3 - 2(-1)^2 - 5(-1)
This equation is not satisfied.
For (0, –5): -5 = (0)^4 + 4(0)^3 - 2(0)^2 - 5(0)
This equation is not satisfied.
For (3, 52): 52 = (3)^4 + 4(3)^3 - 2(3)^2 - 5(3)
This equation is not satisfied.
Option 4 does not satisfy any of the points.
Therefore, the polynomial that has a graph that passes through all the given points is:
y = x^3 - 4x^2 + 2x - 5
To determine which polynomial has a graph that passes through the given points, you can use the method of substitution.
First, substitute the x and y values from each point into each polynomial equation as follows:
1. For the polynomial y = x3 - 4x2 + 2x - 5:
- When x = -3, y = (-3)^3 - 4(-3)^2 + 2(-3) - 5 = 10
- When x = -1, y = (-1)^3 - 4(-1)^2 + 2(-1) - 5 = 0
- When x = 0, y = (0)^3 - 4(0)^2 + 2(0) - 5 = -5
- When x = 3, y = (3)^3 - 4(3)^2 + 2(3) - 5 = 52
2. For the polynomial y = x^3 + 4x^2 - 2x - 5:
- When x = -3, y = (-3)^3 + 4(-3)^2 - 2(-3) - 5 = 28
- When x = -1, y = (-1)^3 + 4(-1)^2 - 2(-1) - 5 = -4
- When x = 0, y = (0)^3 + 4(0)^2 - 2(0) - 5 = -5
- When x = 3, y = (3)^3 + 4(3)^2 - 2(3) - 5 = 43
3. For the polynomial y = x^3 + 4x^2 + 2x - 5:
- When x = -3, y = (-3)^3 + 4(-3)^2 + 2(-3) - 5 = 38
- When x = -1, y = (-1)^3 + 4(-1)^2 + 2(-1) - 5 = -6
- When x = 0, y = (0)^3 + 4(0)^2 + 2(0) - 5 = -5
- When x = 3, y = (3)^3 + 4(3)^2 + 2(3) - 5 = 52
4. For the polynomial y = x^4 + 4x^3 - 2x^2 - 5x:
- When x = -3, y = (-3)^4 + 4(-3)^3 - 2(-3)^2 - 5(-3) = 178
- When x = -1, y = (-1)^4 + 4(-1)^3 - 2(-1)^2 - 5(-1) = 0
- When x = 0, y = (0)^4 + 4(0)^3 - 2(0)^2 - 5(0) = 0
- When x = 3, y = (3)^4 + 4(3)^3 - 2(3)^2 - 5(3) = 268
Based on these calculations, the polynomial that matches all four points is y = x^3 + 4x^2 + 2x - 5. Therefore, the correct answer is:
y = x^3 + 4x^2 + 2x - 5