Question

graph the following points, describe the graph/family and answer each question: (-4, 7) (-2, -5) (0, -9) (2, -5) (4, 7)
what is the family?
what quadrants contain the graph?
is the point (1, -8) a solution to this family? explain.
the graph above should have two x - intercepts. where are they located?
what is the y - intercept of this graph? is it the lowest or highest point on the graph?

Explanation:

Step1: Identify the family of the graph

The points (-4, 7), (-2, -5), (0, -9), (2, -5), (4, 7) are symmetric about the line x = 0. This is the form of a parabola (quadratic - function family). The general form of a quadratic function is \(y=ax^{2}+bx + c\).

Step2: Determine the quadrants

The points (-4, 7) and (4, 7) are in Quadrants II and I respectively. The points (-2, -5) and (2, -5) are in Quadrants III and IV respectively. So the graph contains points in all four quadrants.

Step3: Check if (1, -8) is a solution

Assume the quadratic function \(y = ax^{2}+bx + c\). Using the point (0, -9), we get \(c=-9\). Using (-2, -5) and (2, -5):
For (-2, -5): \(-5=a(-2)^{2}+b(-2)-9\), which simplifies to \(4a-2b=4\) or \(2a - b = 2\).
For (2, -5): \(-5=a(2)^{2}+b(2)-9\), which simplifies to \(4a + 2b=4\) or \(2a + b = 2\).
Solving the system \(

$$\begin{cases}2a - b=2\\2a + b=2\end{cases}$$

\), we add the two equations: \((2a - b)+(2a + b)=2 + 2\), \(4a=4\), \(a = 1\), and \(b = 0\). So the function is \(y=x^{2}-9\).
When \(x = 1\), \(y=1^{2}-9=-8\). So (1, -8) is a solution.

Step4: Find the x - intercepts

Set \(y = 0\) in \(y=x^{2}-9\). Then \(x^{2}-9=0\), which factors to \((x + 3)(x - 3)=0\). So \(x=-3\) and \(x = 3\) are the x - intercepts.

Step5: Find the y - intercept and its position

The y - intercept is the point where \(x = 0\). For \(y=x^{2}-9\), when \(x = 0\), \(y=-9\). Since the coefficient of \(x^{2}\) is positive (\(a = 1>0\)), the parabola opens upwards and the y - intercept \((0,-9)\) is the lowest point on the graph.

Answer:

What is the family?: Quadratic functions
What quadrants contain the graph?: Quadrants I, II, III, IV
Is the point (1, -8) a solution to this family? Explain.: Yes, because for \(y=x^{2}-9\), when \(x = 1\), \(y=-8\)
Where are the x - intercepts located?: \(x=-3\) and \(x = 3\)
What is the y - intercept of this graph? Is it the lowest or highest point on the graph?: The y - intercept is \((0,-9)\) and it is the lowest point on the graph.