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A parabola is a symmetrical shape that you can easily recognize from its U-like curve. In mathematics, parabolas are represented as the graph of a quadratic function, which is an equation of the form \( y = ax^2 + bx + c \). In the given exercise, the equation is \( y = x^2 - 9 \). This indicates it is a basic quadratic equation with no linear term, meaning the parabola opens upwards.The main parts of a parabola include the vertex, which is the highest or lowest point depending on the parabola's orientation, and the axis of symmetry. For the equation \( y = x^2 - 9 \), the vertex is at the point \((0, -9)\), which is the lowest point in this particular graph. The axis of symmetry runs through the vertex, splitting the parabola into two mirror-image halves.

The x-intercepts are the points where the graph of a function crosses the x-axis. This occurs where the y-value is zero. For quadratic equations like \( y = x^2 - 9 \), finding the x-intercepts involves solving the equation \( x^2 - 9 = 0 \). You can achieve this by setting \( y \) to zero and solving for \( x \).In this case:

• Step 1: Set \( y = 0 \), giving you the equation \( x^2 = 9 \).
• Step 2: Solve \( x^2 = 9 \) to find \( x = \pm 3 \).

This solution means the x-intercepts are at the points \((3, 0)\) and \((-3, 0)\), where the parabola intersects the x-axis.

The y-intercept is a specific point where the graph crosses the y-axis. At this point, the x-value is always zero. To find the y-intercept of a quadratic function, simply plug \( x = 0 \) into the equation and solve for \( y \).For the given equation \( y = x^2 - 9 \):

• Substitute \( x = 0 \): \( y = 0^2 - 9 \)
• Thus, \( y = -9 \)

This computation shows that the y-intercept is at the point \((0, -9)\), which is also the vertex of the parabola. Notice how the y-intercept gives insight into the parabola's vertical position on the graph.

Functions can show different kinds of symmetry, but one common type is symmetry about the y-axis. A function is symmetric about the y-axis if flipping it over this axis results in the same graph.For a quadratic function like \( y = x^2 - 9 \), you can check for this symmetry by substituting \(-x\) for \(x\) and checking if the function remains unchanged:

• Substitute \(-x\) into the function: \[ y = (-x)^2 - 9 \]
• Since \((-x)^2 = x^2\), the equation simplifies back to \( y = x^2 - 9 \)

This simplification shows that our function does maintain its form, marking it as symmetric about the y-axis. This symmetry is a helpful tool for sketching graphs, as it allows for easier prediction of the graph's behavior.