The illustration below shows the graph of
\[y\] as a function of
\[x\].
Complete the following sentences based on the graph of the function.
(Enter the
\[x\]-intercepts from least to greatest.)
This is the graph of a

function.
The
\[y\]-intercept of the graph is the function value
\[y=\:\]
.
The
\[x\]-intercepts of the graph (in order from least to greatest) are located at
\[x=\:\]
and
\[x=\:\]
.
The greatest value of
\[y\] is
\[y=\:\]
, and it occurs when
\[x=\:\]
.
For
\[x\] between
\[x=0\] and
\[x=6\], the function value
\[y\:\]

\[\:0\].
A graph of y as a function of x on the coordinate plane.  The horizontal x axis runs from negative 10 to 10 in intervals of 1. The vertical y axis runs from negative 10 to 10 in intervals of 1. The function increases and decreases as it moves left to right. The graph of curved line passes through the points (negative 1, negative 7), (0, 0), (1, 5), (2, 8), (3, 9), (4, 8), (5, 5), (6, 0), and (7, negative 7).

Given the description of the graph and the points it passes through, we can complete the sentences as follows:

1. This is the graph of a **quadratic** function.  
   (Note: This is an approximation based on the symmetry and turning points of the graph. It could also be a polynomial function of a higher degree, but based on the points, quadratic is a reasonable assumption.)

2. The \( y \)-intercept of the graph is the function value \( y = 0 \).

3. The \( x \)-intercepts of the graph (in order from least to greatest) are located at \( x = 0 \) and \( x = 6 \).

4. The greatest value of \( y \) is \( y = 9 \), and it occurs when \( x = 3 \).

5. For \( x \) between \( x = 0 \) and \( x = 6 \), the function value \( y \) **is greater than** \( 0 \).

Would you like to explore any details or have any questions?

Here are 8 related questions:
1. What is the vertex form of the quadratic function based on the given points?
2. How do you find the axis of symmetry for a quadratic function?
3. Can a function have more than two \( x \)-intercepts? Explain why or why not.
4. What is the significance of the \( y \)-intercept in the context of a function?
5. How do you determine the increasing and decreasing intervals of a function from its graph?
6. What are the characteristics of a quadratic function in standard form?
7. How does the value of the leading coefficient affect the direction of a quadratic graph?
8. What are the properties of polynomial functions that distinguish them from other types of functions?

**Tip:** To find the \( x \)-intercepts of a function, set \( y = 0 \) and solve for \( x \).

Explore the graph of a function and analyze its x-intercepts, y-intercept, and maximum value. Understand the characteristics of quadratic functions based on given points. Suitable for Grades 7-9.

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