To determine the path of the ball in terms of transforming the graph of [tex]\( y = x^2 \)[/tex], we need to understand how the quadratic function [tex]\( y = -16x^2 + 32x + 3 \)[/tex] morphs from the standard form. Here are the steps broken down:
1. Standard to General Form:
Start with the standard form of a quadratic function, [tex]\( y = ax^2 + bx + c \)[/tex]. For our function [tex]\( y = -16x^2 + 32x + 3 \)[/tex]:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 32 \)[/tex]
- [tex]\( c = 3 \)[/tex]
2. Significance of Coefficients:
- The coefficient [tex]\( a = -16 \)[/tex] indicates the parabola opens downwards (because [tex]\( a \)[/tex] is negative).
- The coefficients [tex]\( b \)[/tex] and [tex]\( c \)[/tex] help us find the vertex of the parabola, which is crucial for understanding the path.
3. Finding the Vertex:
- The vertex form of a quadratic function is [tex]\( y = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex.
- To find [tex]\( h \)[/tex], use the vertex formula [tex]\( h = -\frac{b}{2a} \)[/tex].
Given [tex]\( a = -16 \)[/tex] and [tex]\( b = 32 \)[/tex]:
[tex]\[ h = -\frac{b}{2a} = -\frac{32}{2(-16)} = 1.0 \][/tex]
4. Determine the Maximum Height (k-value):
- Now substitute [tex]\( h = 1.0 \)[/tex] back into the quadratic function to find [tex]\( k \)[/tex].
[tex]\[ k = a(h)^2 + b(h) + c \][/tex]
Using [tex]\( h = 1.0 \)[/tex]:
[tex]\[ k = -16(1.0)^2 + 32(1.0) + 3 = -16 \cdot 1 + 32 \cdot 1 + 3 = -16 + 32 + 3 = 19.0 \][/tex]
Thus, the vertex [tex]\((h, k)\)[/tex] is [tex]\( (1.0, 19.0) \)[/tex].
5. Graph Transformation:
- The vertex form [tex]\( y = -16(x-1)^2 + 19 \)[/tex] tells us how the graph transforms from the basic [tex]\( y = x^2 \)[/tex]:
- Reflection: [tex]\( y = x^2\)[/tex] opens upwards, while [tex]\( y = -16x^2 \)[/tex] (due to [tex]\( -16 \)[/tex]) reflects it to open downwards.
- Stretching (Vertical Compression): The factor 16 causes a vertical stretch by a factor of 16, making the graph narrower.
- Horizontal Shift: The subtraction inside the squared term [tex]\((x-1)\)[/tex] shifts the graph 1 unit to the right.
- Vertical Shift: The [tex]\(+19\)[/tex] outside the squared term shifts the graph up by 19 units.
Combining these transformations, we get the path of the ball represented by the function [tex]\( y = -16x^2 + 32x + 3 \)[/tex]. The ball reaches its maximum height of 19 feet at [tex]\( x = 1 \)[/tex] second before it starts descending back to the ground.
