We are given the quadratic function $f(x) = 2x^2 + 10x + 11$. Let's go through the steps to answer each part of the question.

(a) Find the vertex:

The vertex form of a quadratic equation is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. But we can find the vertex using the standard form $ax^2 + bx + c$.

To find the x-coordinate of the vertex, we use the formula:

$x = \frac{-b}{2a}$

Here, $a = 2$, $b = 10$, and $c = 11$. Substituting these values into the formula:

$x = \frac{-10}{2(2)} = \frac{-10}{4} = -2.5$

Now, substitute $x = -2.5$ back into the function $f(x)$ to find the y-coordinate of the vertex:

$f(-2.5) = 2(-2.5)^2 + 10(-2.5) + 11$ $f(-2.5) = 2(6.25) - 25 + 11$ $f(-2.5) = 12.5 - 25 + 11 = -1.5$

Thus, the vertex is $(-2.5, -1.5)$.

(b) Find the axis of symmetry:

The axis of symmetry is a vertical line that passes through the x-coordinate of the vertex. From part (a), the x-coordinate of the vertex is $-2.5$. Thus, the axis of symmetry is:

$x = -2.5$

(c) Determine whether there is a maximum or a minimum value and find that value:

Since the coefficient of $x^2$ is positive ($a = 2 > 0$), the parabola opens upwards, meaning it has a minimum value at the vertex.

The minimum value is the y-coordinate of the vertex, which we found to be $-1.5$.

Thus, the minimum value of the function is $-1.5$.

(d) Graph the function:

To graph the function, follow these steps:

1. Plot the vertex $(-2.5, -1.5)$.
2. The axis of symmetry is $x = -2.5$.
3. Identify the y-intercept by evaluating $f(0)$:

$f(0) = 2(0)^2 + 10(0) + 11 = 11$

So, the y-intercept is $(0, 11)$. 4. Reflect the point $(0, 11)$ across the axis of symmetry to get another point $(-5, 11)$. 5. Connect these points to sketch the parabola, making sure it opens upwards and is symmetric about the axis of symmetry.

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Would you like help visualizing this graph or need further details?

Relative questions:

1. How can we convert a quadratic function to its vertex form?
2. What role does the discriminant play in determining the number of x-intercepts of a quadratic function?
3. How does the sign of the coefficient $a$ in a quadratic function affect its graph?
4. What is the relationship between the vertex and the maximum/minimum value of a quadratic function?
5. How can we solve for the roots of $f(x) = 0$ for the given quadratic function?

Tip:

For any quadratic function, the axis of symmetry always passes through the vertex, and its equation is $x = -\frac{b}{2a}$.