Let's solve the problem in the image step by step.

Given the quadratic function $f(x) = x^2 - 2x - 3$, we need to answer the following:

(a) Does the graph of $f$ open up or down?

To determine this, we look at the leading coefficient of $x^2$. In the function $f(x) = x^2 - 2x - 3$, the coefficient of $x^2$ is positive (it is 1). When the leading coefficient is positive, the parabola opens up.

So, the answer is up.

(b) What are the coordinates of the vertex?

The vertex of a parabola given by $f(x) = ax^2 + bx + c$ can be found using the formula for the x-coordinate of the vertex:

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

For the given function $f(x) = x^2 - 2x - 3$, we have:

• $a = 1$
• $b = -2$

Substitute these values into the vertex formula:

$x = \frac{-(-2)}{2(1)} = \frac{2}{2} = 1$

Now, to find the y-coordinate of the vertex, substitute $x = 1$ into the original function $f(x)$:

$f(1) = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4$

Thus, the vertex is $(1, -4)$.

Let me know if you'd like further details on any part, or if you'd like help with the remaining parts.

Here are five related questions to explore:

1. How do you find the axis of symmetry for a quadratic function?
2. What is the significance of the vertex in a parabola?
3. How can you find the x-intercepts of a quadratic function?
4. What is the effect of changing the coefficient of $x^2$ on the graph of a quadratic function?
5. How does the y-intercept relate to the constant term in a quadratic function?

Tip: Always check the sign of the coefficient of $x^2$ first to quickly determine whether a parabola opens upwards or downwards.