Question

pumpkin paths
this bear robot is also launching pumpkins.
the function $b(x) = -0.15(x - 4)^2 + 6.5$ represents the path of the pumpkin.
a. drag the point to change the path.
b. discuss: how did changing the graph affect the equation?

Explanation:

Part b Explanation:

To determine how changing the graph affects the equation \( b(x) = -0.15(x - 4)^2 + 6.5 \), we analyze the vertex form of a quadratic function, \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the direction and width of the parabola.

• Vertex \((h, k)\): The vertex of the original parabola is \((4, 6.5)\) (from \(h = 4\), \(k = 6.5\)). If we drag the vertex (the point \((4, 6.5)\)) horizontally or vertically:
• Moving the vertex left/right changes \(h\) (the \(x\)-coordinate of the vertex), so the term \((x - h)\) in the equation updates (e.g., moving right to \(x = 5\) changes \((x - 4)\) to \((x - 5)\)).
• Moving the vertex up/down changes \(k\) (the \(y\)-coordinate of the vertex), so the constant term \(+6.5\) in the equation updates (e.g., moving up to \(y = 7\) changes \(+6.5\) to \(+7\)).

• Direction/Width (\(a\)): The coefficient \(a = -0.15\) determines if the parabola opens up/down (negative \(a\) opens down) and its width (smaller \(|a|\) means wider, larger \(|a|\) means narrower). If we stretch or compress the parabola (or flip its direction), \(a\) changes:
• Making the parabola wider (flatter) decreases \(|a|\) (e.g., \(a = -0.1\) is wider than \(a = -0.15\)).
• Making it narrower (steeper) increases \(|a|\) (e.g., \(a = -0.2\) is narrower than \(a = -0.15\)).
• Flipping the parabola to open up changes \(a\) to positive (e.g., \(a = 0.15\)).

Part b Answer:

Changing the graph (by dragging the vertex or adjusting the parabola’s shape) affects the equation \( b(x) = -0.15(x - 4)^2 + 6.5 \) as follows:

• Vertex movement (horizontal/vertical): Changes the vertex \((h, k)\) in the vertex form \( y = a(x - h)^2 + k \), so \(h\) (inside \((x - h)\)) or \(k\) (the constant term) updates.
• Shape/width/direction: Changes the coefficient \(a\) (e.g., stretching/compressing changes \(|a|\); flipping direction changes \(a\)’s sign).

For example:

• Dragging the vertex \((4, 6.5)\) right to \(x = 5\) changes the equation to \( b(x) = -0.15(x - 5)^2 + 6.5 \).
• Dragging the vertex up to \(y = 7\) changes it to \( b(x) = -0.15(x - 4)^2 + 7 \).
• Making the parabola wider (flatter) changes \(a\) to a smaller absolute value (e.g., \(a = -0.1\)), so the equation becomes \( b(x) = -0.1(x - 4)^2 + 6.5 \).

(Note: For part a, the action is to drag the purple point \((4, 6.5)\) to adjust the parabola’s vertex or shape, but part b focuses on the relationship between the graph’s change and the equation’s parameters.)

Answer:

Part b Explanation:

To determine how changing the graph affects the equation \( b(x) = -0.15(x - 4)^2 + 6.5 \), we analyze the vertex form of a quadratic function, \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the direction and width of the parabola.

• Vertex \((h, k)\): The vertex of the original parabola is \((4, 6.5)\) (from \(h = 4\), \(k = 6.5\)). If we drag the vertex (the point \((4, 6.5)\)) horizontally or vertically:
• Moving the vertex left/right changes \(h\) (the \(x\)-coordinate of the vertex), so the term \((x - h)\) in the equation updates (e.g., moving right to \(x = 5\) changes \((x - 4)\) to \((x - 5)\)).
• Moving the vertex up/down changes \(k\) (the \(y\)-coordinate of the vertex), so the constant term \(+6.5\) in the equation updates (e.g., moving up to \(y = 7\) changes \(+6.5\) to \(+7\)).

• Direction/Width (\(a\)): The coefficient \(a = -0.15\) determines if the parabola opens up/down (negative \(a\) opens down) and its width (smaller \(|a|\) means wider, larger \(|a|\) means narrower). If we stretch or compress the parabola (or flip its direction), \(a\) changes:
• Making the parabola wider (flatter) decreases \(|a|\) (e.g., \(a = -0.1\) is wider than \(a = -0.15\)).
• Making it narrower (steeper) increases \(|a|\) (e.g., \(a = -0.2\) is narrower than \(a = -0.15\)).
• Flipping the parabola to open up changes \(a\) to positive (e.g., \(a = 0.15\)).

Part b Answer:

Changing the graph (by dragging the vertex or adjusting the parabola’s shape) affects the equation \( b(x) = -0.15(x - 4)^2 + 6.5 \) as follows:

• Vertex movement (horizontal/vertical): Changes the vertex \((h, k)\) in the vertex form \( y = a(x - h)^2 + k \), so \(h\) (inside \((x - h)\)) or \(k\) (the constant term) updates.
• Shape/width/direction: Changes the coefficient \(a\) (e.g., stretching/compressing changes \(|a|\); flipping direction changes \(a\)’s sign).

For example:

• Dragging the vertex \((4, 6.5)\) right to \(x = 5\) changes the equation to \( b(x) = -0.15(x - 5)^2 + 6.5 \).
• Dragging the vertex up to \(y = 7\) changes it to \( b(x) = -0.15(x - 4)^2 + 7 \).
• Making the parabola wider (flatter) changes \(a\) to a smaller absolute value (e.g., \(a = -0.1\)), so the equation becomes \( b(x) = -0.1(x - 4)^2 + 6.5 \).

(Note: For part a, the action is to drag the purple point \((4, 6.5)\) to adjust the parabola’s vertex or shape, but part b focuses on the relationship between the graph’s change and the equation’s parameters.)