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.