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.