To create a quadratic model for the given dataset, we can assume it has the form:
\[ y = ax^2 + bx + c \]
Given your data points:
- \( (-1, 6) \)
- \( (0, 5) \)
- \( (1, 6) \)
- \( (2, 9) \)
- \( (3, 14) \)
We can set up a system of equations using some of these points to solve for \(a\), \(b\), and \(c\).
Using the points (0, 5), (1, 6), and (2, 9):
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From the point (0, 5): \[ 5 = a(0)^2 + b(0) + c \implies c = 5 \]
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From the point (1, 6): \[ 6 = a(1)^2 + b(1) + 5 \implies a + b + 5 = 6 \implies a + b = 1 \quad (1) \]
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From the point (2, 9): \[ 9 = a(2)^2 + b(2) + 5 \implies 4a + 2b + 5 = 9 \implies 4a + 2b = 4 \implies 2a + b = 2 \quad (2) \]
Now we have a system of linear equations:
\[ \begin{align*}
- & \quad a + b = 1 \
- & \quad 2a + b = 2 \end{align*} \]
Step 1: Solve the system of equations
Subtract equation (1) from equation (2):
\[ (2a + b) - (a + b) = 2 - 1 \implies 2a - a = 1 \implies a = 1 \]
Now substitute \(a = 1\) back into equation (1):
\[ 1 + b = 1 \implies b = 0 \]
So we have:
\[ a = 1, \quad b = 0, \quad c = 5 \]
Final model
The quadratic model is:
\[ y = 1x^2 + 0x + 5 \implies y = x^2 + 5 \]
Step 2: Create XY chart
To plot the XY chart with the quadratic equation \(y = x^2 + 5\) and the given points:
- At \( x = -1: y = (-1)^2 + 5 = 1 + 5 = 6\) → point (-1, 6)
- At \( x = 0: y = (0)^2 + 5 = 5\) → point (0, 5)
- At \( x = 1: y = (1)^2 + 5 = 1 + 5 = 6\) → point (1, 6)
- At \( x = 2: y = (2)^2 + 5 = 4 + 5 = 9\) → point (2, 9)
- At \( x = 3: y = (3)^2 + 5 = 9 + 5 = 14\) → point (3, 14)
Graph
You can plot these points on a chart, and also include the curve from the quadratic model. The curve will be a parabola opening upwards centered around the y-coordinate of 5.
For an accurate graphical representation, you would typically use graphing software or tools like Desmos, Excel, or Python's matplotlib, but you can imagine a smooth curve passing through these points that depicts \(y = x^2 + 5\).