To graph the quadratic function \( y = 2x^2 - 3x + 4 \), follow these steps:

### 1. Identify the Quadratic Function Form:
The equation is in the standard form of a quadratic function:
\[ y = ax^2 + bx + c \]
Here, \( a = 2 \), \( b = -3 \), and \( c = 4 \).

### 2. Determine the Vertex:
The vertex of a parabola is a crucial point, as it represents the highest or lowest point of the graph depending on whether the parabola opens upward (when \( a > 0 \)) or downward (when \( a < 0 \)). The vertex can be found using the formula:
\[ x = -\frac{b}{2a} \]
Substitute the values of \( a \) and \( b \):
\[ x = -\frac{-3}{2(2)} = \frac{3}{4} \]

Now, find the corresponding \( y \)-value by substituting \( x = \frac{3}{4} \) back into the equation:
\[ y = 2\left(\frac{3}{4}\right)^2 - 3\left(\frac{3}{4}\right) + 4 \]
\[ y = 2\left(\frac{9}{16}\right) - \frac{9}{4} + 4 \]
\[ y = \frac{18}{16} - \frac{36}{16} + \frac{64}{16} \]
\[ y = \frac{46}{16} = \frac{23}{8} = 2.875 \]
So, the vertex is \( \left(\frac{3}{4}, \frac{23}{8}\right) \).

### 3. Determine the Axis of Symmetry:
The axis of symmetry is a vertical line that passes through the vertex, given by the equation:
\[ x = \frac{3}{4} \]

### 4. Find the Y-intercept:
The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation:
\[ y = 2(0)^2 - 3(0) + 4 = 4 \]
So, the y-intercept is \( (0, 4) \).

### 5. Find Additional Points:
Choose a couple of other x-values to find additional points. For example:
- For \( x = 1 \):
  \[ y = 2(1)^2 - 3(1) + 4 = 2 - 3 + 4 = 3 \]
  Point: \( (1, 3) \)
  
- For \( x = -1 \):
  \[ y = 2(-1)^2 - 3(-1) + 4 = 2 + 3 + 4 = 9 \]
  Point: \( (-1, 9) \)

### 6. Plot the Points and Draw the Parabola:
- Plot the vertex \( \left(\frac{3}{4}, \frac{23}{8}\right) \), the y-intercept \( (0, 4) \), and the additional points \( (1, 3) \) and \( (-1, 9) \).
- Draw the axis of symmetry \( x = \frac{3}{4} \).
- Connect the points with a smooth, U-shaped curve that opens upwards, since \( a = 2 > 0 \).

### 7. Analyze the Graph:
- The parabola opens upwards.
- The vertex \( \left(\frac{3}{4}, \frac{23}{8}\right) \) is the minimum point.
- The y-intercept is \( (0, 4) \), and the graph will continue to rise on either side of the vertex.

Do you want more details or have any questions? 

Here are some related questions:

1. How do you find the x-intercepts of a quadratic function?
2. What is the significance of the vertex in the graph of a parabola?
3. How does the value of \( a \) affect the shape of the parabola?
4. What are the steps to complete the square for a quadratic function?
5. How do you graph a quadratic function that opens downward?
6. What is the relationship between the discriminant and the number of x-intercepts?
7. How do transformations affect the graph of a quadratic function?
8. What are real-world applications of quadratic functions?

**Tip:** When graphing quadratic functions, always start by identifying the vertex and axis of symmetry; these will guide the rest of the graph.

Learn how to graph the quadratic function y = 2x^2 - 3x + 4 step-by-step. Understand key concepts such as finding the vertex, axis of symmetry, y-intercept, and plotting additional points. Suitable for students in Grades 9-12.

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