To analyze the quadratic function [tex]\( y = 4x^2 - 8x + 5 \)[/tex] and extract the necessary features such as the vertex, axis of symmetry, [tex]\( x \)[/tex]-intercepts, and [tex]\( y \)[/tex]-intercept, follow these detailed steps:
1. Vertex:
The vertex form of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] allows us to find the vertex using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
Given the quadratic function [tex]\( y = 4x^2 - 8x + 5 \)[/tex]:
- Coefficient [tex]\( a = 4 \)[/tex]
- Coefficient [tex]\( b = -8 \)[/tex]
- Coefficient [tex]\( c = 5 \)[/tex]
Calculate [tex]\( x \)[/tex] of the vertex:
[tex]\[
x = -\frac{-8}{2 \cdot 4} = \frac{8}{8} = 1
\][/tex]
Substitute [tex]\( x = 1 \)[/tex] back into the quadratic function to find [tex]\( y \)[/tex]:
[tex]\[
y = 4(1)^2 - 8(1) + 5 = 4 - 8 + 5 = 1
\][/tex]
Therefore, the vertex is [tex]\( (1, 1) \)[/tex].
2. Axis of Symmetry:
The axis of symmetry of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the vertex. For our function, it is [tex]\( x = 1 \)[/tex].
3. [tex]\( x \)[/tex]-intercepts:
[tex]\( x \)[/tex]-intercepts occur where the quadratic function equals zero, i.e., solving [tex]\( 4x^2 - 8x + 5 = 0 \)[/tex].
Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = (-8)^2 - 4 \cdot 4 \cdot 5 = 64 - 80 = -16
\][/tex]
Since the discriminant is negative ([tex]\( -16 \)[/tex]), the quadratic equation has no real roots. Thus, there are no real [tex]\( x \)[/tex]-intercepts.
4. [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept is found by setting [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 4(0)^2 - 8(0) + 5 = 5
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is [tex]\( 5 \)[/tex], or the point [tex]\( (0, 5) \)[/tex].
Summarizing the important features:
- Vertex: [tex]\((1, 1)\)[/tex]
- Axis of Symmetry: [tex]\( x = 1 \)[/tex]
- [tex]\( x \)[/tex]-intercepts: None (no real roots)
- [tex]\( y \)[/tex]-intercept: [tex]\( 5 \)[/tex]
These details help in graphing the parabola accurately.
