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A quadratic equation is one of the most fundamental concepts in algebra. It takes the general form \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In this article, however, we're looking at a specific case: the equation \( y = 1 - x^2 \). This is a simple equation where \( a = -1 \), \( b = 0 \), and \( c = 1 \).
The unique aspect of quadratic equations is their ability to form parabolas when graphed on a coordinate plane. The parabola's orientation—whether it opens upwards or downwards—depends on the sign of \( a \). Since the coefficient of \( x^2 \) in our equation is negative, the parabola opens downward. This feature will determine the overall shape when you plot points on a graph and connect them to visualize the curve.

X-intercepts are the points where the graph of an equation crosses the x-axis. For these points, the value of \( y \) is always zero. To find the x-intercepts, you set the equation equal to zero and solve for \( x \).
For our equation \( y = 1 - x^2 \), setting \( y = 0 \) results in:

• \( 1 - x^2 = 0 \)
• Solving, we get \( x^2 = 1 \)
• Thus, \( x = \pm 1 \)

The x-intercepts are the points \((1, 0)\) and \((-1, 0)\). These intercepts are critical in sketching the graph because they represent the points where the curve crosses the x-axis, visually confirming the equation's real-number solutions.

The y-intercept is where the graph of an equation crosses the y-axis. For these points, \( x \) is always zero. To determine the y-intercept, substitute \( x = 0 \) into the equation and solve for \( y \).
In our scenario with \( y = 1 - x^2 \):

• Substituting \( x = 0 \) gives \( y = 1 - 0^2 = 1 \)

Therefore, the y-intercept is the point \((0, 1)\). This is significant because it provides a starting point on the y-axis from which you can graph the curve. Every line must intercept the y-axis, making this point a key reference in plotting your graph.

Symmetry in a parabola is a powerful property, making the graph easier to analyze and draw. A parabola can exhibit symmetry about the y-axis, the x-axis, or the origin.
For the equation \( y = 1 - x^2 \), the parabola shows symmetry about the y-axis. This means if you were to fold the graph along the y-axis, both sides would match perfectly. To test for y-axis symmetry, replace \( x \) with \( -x \) in the equation, and see if it remains unchanged:

• \( y = 1 - (-x)^2 \) simplifies to \( y = 1 - x^2 \)

Since the outcome matches the original equation, y-axis symmetry is confirmed. You can't have symmetry about the x-axis because that would mean \( y = -y \), a contradiction with this parabola's equation. Symmetry about the origin isn't applicable here either as the transformation \( -y = 1 - x^2 \) doesn't revert back to the original equation. Recognizing symmetry helps efficiently plot the graph and understand its properties.