what is the graph of the function y = 2x^2-12x+10?

To graph the function \( y = 2x^2 - 12x + 10 \), we need to identify key features of the parabola, such as its vertex, axis of symmetry, and the y-intercept.

### 1. **Identify the Vertex:**
The function is in the form \( y = ax^2 + bx + c \), where \( a = 2 \), \( b = -12 \), and \( c = 10 \). The x-coordinate of the vertex can be found using the formula:

\[
x = -\frac{b}{2a}
\]

Substituting the values:

\[
x = -\frac{-12}{2(2)} = \frac{12}{4} = 3
\]

Now, substitute \( x = 3 \) back into the equation to find the y-coordinate of the vertex:

\[
y = 2(3)^2 - 12(3) + 10 = 2(9) - 36 + 10 = 18 - 36 + 10 = -8
\]

So, the vertex is at \( (3, -8) \).

### 2. **Axis of Symmetry:**
The axis of symmetry is the vertical line that passes through the vertex:

\[
x = 3
\]

### 3. **Y-Intercept:**
The y-intercept is found by setting \( x = 0 \):

\[
y = 2(0)^2 - 12(0) + 10 = 10
\]

So, the y-intercept is \( (0, 10) \).

### 4. **Plotting the Graph:**
- The parabola opens upwards since \( a = 2 \) is positive.
- The vertex is at \( (3, -8) \).
- The y-intercept is at \( (0, 10) \).
- The axis of symmetry is \( x = 3 \).

The graph will be a parabola opening upwards with the lowest point at \( (3, -8) \), crossing the y-axis at \( (0, 10) \).

### Summary of Key Points:
- **Vertex:** \( (3, -8) \)
- **Axis of Symmetry:** \( x = 3 \)
- **Y-Intercept:** \( (0, 10) \)
- **Opening:** Upwards

If you'd like, I can provide a sketch of the graph. Would you like more details or have any questions?

### Related Questions:
1. How do you find the x-intercepts of the parabola?
2. What is the significance of the vertex in the graph of a parabola?
3. How do you determine if a parabola opens upwards or downwards?
4. What is the effect of changing the coefficient \( a \) on the graph of the parabola?
5. How does the graph change if you alter the constant term \( c \)?

### Tip:
When finding the vertex of a quadratic function, the formula \( x = -\frac{b}{2a} \) is a quick way to determine the line of symmetry and the vertex's x-coordinate.

Learn how to graph the function y = 2x^2 - 12x + 10 step-by-step. Understand the vertex, axis of symmetry, and y-intercept of the parabola. Suitable for students in Grades 9-12.

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