To find the requested characteristics from a graph of a quadratic function, let's analyze them step by step using the provided points, $(0, 4)$ and $(-4, 0)$. These points help us gather information about the vertex, axis of symmetry, intercepts, and domain and range.

a) Vertex:

The vertex of a parabola is the point where the function reaches its maximum or minimum. Since you didn't provide the graph, I'll infer based on the points that the vertex is likely in between or near $x = -2$, given the symmetry of quadratic functions and the point $(0, 4)$. For a more precise answer, I would need more points or the full equation of the function.

If the vertex is $(h, k)$, it could be located near the midpoint of the x-values of the points provided. The point $(0, 4)$ suggests the parabola opens downwards if this point is a maximum.

Approximate vertex: $(-2, k)$ based on symmetry.

b) Axis of symmetry:

The axis of symmetry is a vertical line that passes through the vertex. Based on the vertex location, the axis of symmetry would be at $x = -2$.

Axis of symmetry: $x = -2$

c) y-intercept:

The y-intercept occurs when $x = 0$. From the provided point $(0, 4)$, we already know the y-intercept.

y-intercept: $(0, 4)$

d) x-intercept(s):

The x-intercepts are the points where the graph crosses the x-axis, i.e., where $y = 0$. One of the points given is $(-4, 0)$, so one of the x-intercepts is $(-4, 0)$. There may be another x-intercept, but without more details from the graph, we can only state that $(-4, 0)$ is one intercept.

x-intercept: $(-4, 0)$

e) Domain and range:

• The domain of a quadratic function is always all real numbers, i.e., $(-\infty, \infty)$.
• The range depends on whether the parabola opens upwards or downwards. Since we have a maximum point at $(0, 4)$, the parabola likely opens downward, meaning the range will be from the maximum value (at the vertex) downward to negative infinity.

Domain: $(-\infty, \infty)$
Range: $(-\infty, 4]$

Would you like more details, or do you have any questions?

Here are 5 related questions for further exploration:

1. How can you calculate the exact vertex of a quadratic function from its equation?
2. How does the axis of symmetry relate to the roots of the quadratic equation?
3. What are other methods to find x-intercepts without a graph?
4. How would the domain and range change if the parabola opened upwards?
5. Can a quadratic function have more than two x-intercepts?

Tip: The vertex formula for a quadratic function $f(x) = ax^2 + bx + c$ is $x = -\frac{b}{2a}$.