I got a quadratic graph with a point at (0,-3)(-4,-4)(-2,-2)

Question

Creative creation for this function shown on the graph

Answer 1

To create a quadratic function that fits the points (0, -3), (-4, -4), and (-2, -2), we can express the quadratic function in the standard form:

\[ f(x) = ax^2 + bx + c \]

Step 1: Set Up the System of Equations

Using the points given, we can create a system of equations.

1. Using the point \((0, -3)\): \[ f(0) = a(0)^2 + b(0) + c = c = -3 \]

2. Using the point \((-4, -4)\): \[ f(-4) = a(-4)^2 + b(-4) + c = 16a - 4b - 3 = -4 \] \[ 16a - 4b = -1 \quad \text{(Equation 1)} \]

3. Using the point \((-2, -2)\): \[ f(-2) = a(-2)^2 + b(-2) + c = 4a - 2b - 3 = -2 \] \[ 4a - 2b = 1 \quad \text{(Equation 2)} \]

Step 2: Solve the System of Equations

Now we have the following equations to solve:

1. \( 16a - 4b = -1 \)
2. \( 4a - 2b = 1 \)

From Equation 2, we can simplify it:

\[ 2a - b = 0.5 \quad \text{(Equation 3)} \] \[ b = 2a - 0.5 \]

Substituting Equation 3 into Equation 1:

\[ 16a - 4(2a - 0.5) = -1 \]

Expanding this gives:

\[ 16a - 8a + 2 = -1 \] \[ 8a + 2 = -1 \] \[ 8a = -3 \] \[ a = -\frac{3}{8} \]

Now using \(a\) to find \(b\) using Equation 3:

\[ b = 2\left(-\frac{3}{8}\right) - 0.5 = -\frac{6}{8} - \frac{4}{8} = -\frac{10}{8} = -\frac{5}{4} \]

Now we have \(a\) and \(b\). We know \(c = -3\).

Step 3: Formulate the Polynomial

Putting it all together, we get:

\[ f(x) = -\frac{3}{8}x^2 - \frac{5}{4}x - 3 \]

Step 4: Creative Description

This quadratic function represents a gentle downward-opening parabola that gracefully touches the y-axis at the point (-3) and passes through the points (-4, -4) and (-2, -2). The curvature suggests a gradual decline, reminiscent of a hillside that slopes inward towards a hidden treasure at \(y = -3\). This treasure lies at the vertex of the parabola, emphasizing the balance between ascent and descent in this mathematical landscape.

Step 5: Visualizing the Graph

Plotting the function will show:

• The y-intercept at (0, -3).
• Another point where the graph touches (-4, -4) and (-2, -2), establishing a beautiful arc.
• The function characterizes a quadratic curve indicative of real-world motion, such as the trajectory of an object under the influence of gravity, where the points represent key positions in its descent.

This unique quadratic showcases the beauty of mathematics intertwined with creative thought, manifesting functional elegance and geometry beautifully on the graph.