To understand the path of the ball described by the quadratic function [tex]\( y = -16x^2 + 32x + 3 \)[/tex] in terms of transformations of the graph of [tex]\( y = x^2 \)[/tex], let's break down the steps in detail.
### Step 1: Standard Form and Coefficients
The given quadratic function is in the form [tex]\( y = ax^2 + bx + c \)[/tex] where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 32 \)[/tex]
- [tex]\( c = 3 \)[/tex]
### Step 2: Identify Initial Transformations
Compare the given function [tex]\( y = -16x^2 + 32x + 3 \)[/tex] to the basic parabola [tex]\( y = x^2 \)[/tex]:
- The coefficient [tex]\( -16 \)[/tex] indicates a vertical stretch and reflection over the x-axis.
- The term [tex]\( 32x \)[/tex] suggests a horizontal shift.
- The constant term [tex]\( 3 \)[/tex] indicates a vertical shift upwards by 3 units.
### Step 3: Complete the Square
To better understand the horizontal and vertical shifts, complete the square to transform the quadratic function into vertex form [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex:
1. Start with the quadratic function:
[tex]\[
y = -16x^2 + 32x + 3
\][/tex]
2. Factor out [tex]\( -16 \)[/tex] from the quadratic terms:
[tex]\[
y = -16(x^2 - 2x) + 3
\][/tex]
3. Complete the square inside the parenthesis:
- Take half of the coefficient of [tex]\( x \)[/tex] inside the parenthesis (which is [tex]\(-2\)[/tex]), square it to get 1, and add and subtract this value inside the parenthesis:
[tex]\[
y = -16(x^2 - 2x + 1 - 1) + 3
\][/tex]
[tex]\[
y = -16((x - 1)^2 - 1) + 3
\][/tex]
4. Simplify the expression:
[tex]\[
y = -16(x - 1)^2 + 16 + 3
\][/tex]
[tex]\[
y = -16(x - 1)^2 + 19
\][/tex]
The quadratic function is now in vertex form:
[tex]\[
y = -16(x - 1)^2 + 19
\][/tex]
### Step 4: Interpret the Transformations
The vertex form [tex]\( y = -16(x - 1)^2 + 19 \)[/tex] reveals the following transformations of the graph [tex]\( y = x^2 \)[/tex]:
1. Reflection and Vertical Stretch: The term [tex]\(-16(x - 1)^2\)[/tex] shows a reflection across the x-axis and a vertical stretch by a factor of 16.
2. Horizontal Shift: The [tex]\((x - 1)\)[/tex] term indicates the graph is shifted to the right by 1 unit.
3. Vertical Shift: The +19 indicates the graph is shifted upwards by 19 units.
### Conclusion
The path of the ball, given by the quadratic function [tex]\( y = -16x^2 + 32x + 3 \)[/tex], can be described in terms of transformations starting from the graph of [tex]\( y = x^2 \)[/tex]:
- Reflect [tex]\( y = x^2 \)[/tex] across the x-axis to get [tex]\( y = -x^2 \)[/tex].
- Stretch the reflected graph vertically by a factor of 16 to get [tex]\( y = -16x^2 \)[/tex].
- Shift the graph horizontally to the right by 1 unit.
- Finally, shift the graph vertically upwards by 19 units.
Thus, the equation [tex]\( y = -16(x - 1)^2 + 19 \)[/tex] represents the transformed path of the ball.
