functions for this equation g(x) = 2x2 – 4x + 5

To explore the functions related to the equation \( g(x) = 2x^2 - 4x + 5 \), we can look into various aspects of the function. Here are some key points we can analyze:

1. **Vertex**: The vertex of the parabola represented by \( g(x) \).
2. **Axis of Symmetry**: The vertical line that passes through the vertex.
3. **Roots**: The x-values where \( g(x) = 0 \) (if any).
4. **Y-intercept**: The y-value where \( x = 0 \).
5. **End Behavior**: The behavior of the function as \( x \) approaches \( \pm \infty \).
6. **Increasing/Decreasing Intervals**: Where the function is increasing or decreasing.
7. **First Derivative**: To find the critical points and analyze the slope.
8. **Second Derivative**: To find the concavity and inflection points.

Let's go through these points in detail:

1. **Vertex**:
   The vertex of a quadratic function \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \).
   Here, \( a = 2 \) and \( b = -4 \):
   \[
   x = -\frac{-4}{2 \cdot 2} = \frac{4}{4} = 1
   \]
   Substituting \( x = 1 \) back into the equation to find \( y \):
   \[
   g(1) = 2(1)^2 - 4(1) + 5 = 2 - 4 + 5 = 3
   \]
   So, the vertex is at \( (1, 3) \).

2. **Axis of Symmetry**:
   The axis of symmetry is the vertical line through the vertex:
   \[
   x = 1
   \]

3. **Roots**:
   To find the roots, we solve \( 2x^2 - 4x + 5 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
   \[
   a = 2, \; b = -4, \; c = 5
   \]
   \[
   x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(5)}}{2(2)}
   \]
   \[
   x = \frac{4 \pm \sqrt{16 - 40}}{4}
   \]
   \[
   x = \frac{4 \pm \sqrt{-24}}{4}
   \]
   \[
   x = \frac{4 \pm 2i\sqrt{6}}{4}
   \]
   \[
   x = 1 \pm \frac{i\sqrt{6}}{2}
   \]
   The roots are complex: \( 1 \pm \frac{i\sqrt{6}}{2} \).

4. **Y-intercept**:
   The y-intercept is found by setting \( x = 0 \):
   \[
   g(0) = 2(0)^2 - 4(0) + 5 = 5
   \]
   So, the y-intercept is \( (0, 5) \).

5. **End Behavior**:
   As \( x \) approaches \( \pm \infty \), the \( x^2 \) term dominates:
   \[
   \lim_{x \to \pm \infty} g(x) = \lim_{x \to \pm \infty} (2x^2 - 4x + 5) \approx \lim_{x \to \pm \infty} 2x^2 = \pm \infty
   \]
   The parabola opens upwards since \( a = 2 > 0 \).

6. **Increasing/Decreasing Intervals**:
   We can use the first derivative to find these intervals:
   \[
   g'(x) = \frac{d}{dx}(2x^2 - 4x + 5) = 4x - 4
   \]
   Set the first derivative equal to zero to find critical points:
   \[
   4x - 4 = 0 \implies x = 1
   \]
   The function is decreasing on \( (-\infty, 1) \) and increasing on \( (1, \infty) \).

7. **First Derivative**:
   As calculated above:
   \[
   g'(x) = 4x - 4
   \]

8. **Second Derivative**:
   To determine concavity:
   \[
   g''(x) = \frac{d}{dx}(4x - 4) = 4
   \]
   Since \( g''(x) = 4 > 0 \), the function is concave up everywhere.

Would you like further details or explanations on any of these points?

Here are 8 related questions for further exploration:

1. What is the significance of the vertex in a quadratic function?
2. How do you determine the direction in which a parabola opens?
3. What is the importance of the axis of symmetry in a quadratic function?
4. How can you verify if the roots of a quadratic equation are real or complex?
5. What does the y-intercept represent in a function?
6. How does the end behavior of a function help in graphing it?
7. Why is it useful to know where a function is increasing or decreasing?
8. How can the first and second derivatives help in sketching the graph of a function?

**Tip:** When analyzing quadratic functions, always start by identifying the coefficients \( a \), \( b \), and \( c \), as they provide valuable information about the function's properties and behavior.

Discover detailed analysis of the quadratic function g(x) = 2x^2 - 4x + 5. Learn about its vertex, axis of symmetry, roots, y-intercept, end behavior, increasing/decreasing intervals, derivatives, and concavity. Suitable for advanced high school students.

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