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The y-intercept of a function is the point where the graph of the function crosses the y-axis. For any function, to find the y-intercept, we simply set the variable \( x \) to zero and then solve for \( f(x) \). This is because the y-axis is defined by the condition \( x = 0 \).

For the quadratic function \( f(x) = 9 - x^2 \), we find the y-intercept by evaluating the function at \( x = 0 \). Substituting \( x = 0 \) into the function, we calculate:

• \( f(0) = 9 - (0)^2 = 9 \)

Thus, the y-intercept is the point (0, 9) on the coordinate plane. This means the graph of the function touches the y-axis at \( y = 9 \).

The y-intercept provides valuable information about the graph's vertical position relative to the origin, helping us understand the graph's behavior as it crosses the y-axis.

The x-intercepts of a function are points where the graph crosses the x-axis, meaning the function's output is zero at these points. To determine the x-intercepts, you set \( f(x) \) to zero and solve the resulting equation for \( x \).

For the quadratic function \( f(x) = 9 - x^2 \), we find the x-intercepts by solving the equation:

• \( 9 - x^2 = 0 \)

Rearranging gives:

• \( x^2 = 9 \)

By taking the square root of both sides and considering both positive and negative solutions, we find:

• \( x = \pm 3 \)

This means that the x-intercepts are at the points (-3, 0) and (3, 0).

Understanding where a graph intersects the x-axis is crucial for sketching the graph and analyzing its roots or zeros. These points indicate the values of \( x \) where the output of the function is zero.

A quadratic function is a type of polynomial function characterized by its degree of two. The general form of a quadratic function is:

• \( f(x) = ax^2 + bx + c \)

where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These functions create parabolic graphs that can open upwards or downwards depending on the sign of the leading coefficient \( a \).

In the function \( f(x) = 9 - x^2 \), the quadratic term is \( -x^2 \). This negative sign indicates that the parabola opens downwards. The absence of the linear term (\( bx \)) simplifies the problem by focusing only on the quadratic and constant terms.

Key features of quadratic functions include:

• Vertex: The highest or lowest point of the parabola, which can be found using vertex form or by completing the square.
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetric parts. For \( f(x) = ax^2 + bx + c \), the axis is \( x = -\frac{b}{2a} \).
• Intercepts: Where the graph crosses the axes, already discussed in previous sections.

Understanding these features helps in graphing quadratic functions and solving equations related to them.