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Intercepts are key points where a graph intersects the x-axis or y-axis.
They help us understand the behavior and position of a quadratic function.
For a quadratic function like \(f(x) = 16 - x^2\), we find intercepts by solving for specific values of x and f(x).

To find the **y-intercept**, substitute 0 for x in the equation.
In our example: \(x = 0\), then \(f(x) = 16 - (0)^2 = 16\).
So, the y-intercept is (0, 16).

For the **x-intercepts**, set \(f(x) = 0\) and solve for x.
In this case: \(0 = 16 - x^2\). Rearrange it to \(x^2 = 16\).
Taking the square root gives \(x = \pm \sqrt{16}\), which simplifies to \(\textstyle x = 4 \) and \(\textstyle x = -4 \).
So, the x-intercepts are (4, 0) and (-4, 0).

Quadratic equations take the form \(ax^2 + bx + c = 0 \). They produce a graph with a U-shape called a parabola.
These equations can have a maximum of two real solutions.

A key feature is the vertex, which is the highest or lowest point on the graph.
The direction of the parabola (opening upwards or downwards) depends on the sign of the leading coefficient (a).
If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards.

Intercepts are essential in understanding the parabola's position.
The y-intercept occurs where the graph crosses the y-axis, and the x-intercepts occur where the graph crosses the x-axis.
These intercepts are derived directly from the quadratic formula.
For example, for \(f(x) = 16 - x^2\), it is a downward opening parabola because our \(a\) is -1.

Analyzing the graph of a quadratic function provides insights into its properties and behavior.
Start by identifying the intercepts, which provide key points of intersection with the axes.

Next, determine the vertex and the axis of symmetry.
The vertex can be found using the formula \(x = -\frac{b}{2a}\).
In our function \(f(x) = 16 - x^2\), there is no linear term (b=0), so the vertex is at x=0.
The vertex here is (0,16).
The axis of symmetry is the vertical line passing through the vertex.

Plot important points, including the vertex and intercepts, to sketch the graph.
Ensure you understand how the parabola opens (upwards or downwards).
For \(f(x) = 16 - x^2\), the parabola opens downward.
These steps give a comprehensive view of the quadratic function's behavior.