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A quadratic function is a type of polynomial with a degree of 2. It is usually expressed in the standard form as \( f(x) = ax^2 + bx + c \). The graph of a quadratic function is a curve called a parabola. The values \( a \), \( b \), and \( c \) are coefficients that determine the shape and position of the parabola in a coordinate plane.
For the given function, \( f(x) = -2x^2 + 1 \), the coefficients are \( a = -2 \), \( b = 0 \), and \( c = 1 \).
This specific format shows that our parabola will be vertically in place. If the value of \( a \) were multiplied by a negative number, our parabola opens downward. Conversely, if \( a \) were positive, it would opened upward.

• \( a \) affects the width and the direction of the parabola.
• When \( b \) is zero, the vertex is on the y-axis.
• The value of \( c \) represents the y-intercept.

The vertex of a parabola is its highest or lowest point, depending on its opening direction. For quadratic functions in standard form, you can find the vertex using the formula: \( x = -\frac{b}{2a} \). This formula gives the x-coordinate of the vertex. Once you have the x-coordinate, substitute it back into the function to find the y-coordinate.
For \( f(x) = -2x^2 + 1 \), with \( b = 0 \), the calculation simplifies to \( x = 0 \). Plug \( x = 0 \) back into the function to get the y-coordinate, resulting in \( f(0) = 1 \).
Therefore, the vertex of this parabola is \((0, 1)\). Because \( a = -2 \) is negative, this vertex is the maximum point of the parabola. It appears at the top of our downward opening curve.

The axis of symmetry is a vertical line that splits the parabola into two mirror-image halves. This is fundamental in understanding symmetry in quadratic functions as it provides a reference line for sketching the parabola.
The equation for the axis of symmetry in a quadratic function in standard form is \( x = -\frac{b}{2a} \). Similar to finding the vertex, for our function \( f(x) = -2x^2 + 1 \), with \( b = 0 \), the axis of symmetry is \( x = 0 \).

• This tells us that the parabola is symmetric about the line \( x = 0 \).
• Every point on the parabola one side of this line has an exactly mirrored point on the opposite side.

Knowing this helps when plotting additional points on the graph, as you can easily find mirrored points.

The y-intercept of a graph is where the graph intersects the y-axis. For quadratic functions, it exists when \( x = 0 \). At this point, the function simplifies to \( f(x) = c \) because all \( x \)-terms become zero.
For \( f(x) = -2x^2 + 1 \), the y-intercept is obtained by setting \( x = 0 \). The resulting value is \( f(0) = 1 \).

• Therefore, the y-intercept of our function is at the point \((0, 1)\), which is also the vertex in this case.
• This point gives us an essential reference for sketching the parabola as it provides a specific starting point.

Understanding the y-intercept is vital because it directly tells us where the graph crosses the y-axis, offering a tangible connection of the function graph to the coordinate plane. Knowing it coincides with the vertex here simplifies its plotting significantly.