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The vertex of a quadratic function is a key feature, as it represents the highest or lowest point on the graph, depending on the direction the parabola opens. In this example, the quadratic function given is \( f(x) = -x^2 + 2x + 3 \). To find the vertex, we need to use the vertex formula, which identifies the x-coordinate as \( h = -\frac{b}{2a} \). Here, \( a = -1 \) and \( b = 2 \), resulting in \( h = 1 \).

Once the x-coordinate is determined, substitute it back into the function to solve for the y-coordinate of the vertex: \( k = f(1) = -1 + 2 + 3 = 4 \). Therefore, the vertex of the parabola is at the point \( (1, 4) \).

This vertex is crucial, as it indicates a maximum point for the function since the parabola opens downward (because \( a < 0 \)). By understanding the vertex, you can determine both the direction and the extreme value of the parabola.

The x-intercepts of a quadratic are the points where the graph crosses the x-axis, so the y-value is zero at these points. For our given function \( f(x) = (3-x)(x+1) \), you can find these intercepts by setting the function equal to zero: \( (3-x)(x+1) = 0 \). This format helps us immediately spot two solutions for \( x \).

* First, set each factor equal to zero:

• \( 3-x=0 \) simplifies to \( x=3 \)
• \( x+1=0 \) simplifies to \( x=-1 \)

As a result, the x-intercepts are at points \((3, 0)\) and \((-1, 0)\).

By identifying x-intercepts, you can easily mark where the parabola intersects the x-axis on the graph. This is particularly useful for visualizing the shape and positioning of the quadratic curve.

The y-intercept is the point where the graph crosses the y-axis, and at this point, \( x \) is zero. To find the y-intercept of our function, substitute \( x = 0 \) into the original function: \( f(0) = (3 - 0)(0 + 1) \).

Calculate to get \( f(0) = 3 \, \cdot \, 1 = 3 \). Thus, the y-intercept of the function is the point \((0, 3)\).

The y-intercept provides a starting point for sketching the graph. It marks where the parabola crosses the y-axis, giving you a reference point from which to plot the rest of the curve. By knowing the intercepts, you can align your graph correctly with both the x and y axes.

Graphing a quadratic function, such as a parabola, involves plotting specific key points, including intercepts and the vertex. With our function \( f(x) = -x^2 + 2x + 3 \), we have identified critical points:

• X-Intercepts: \((3, 0)\) and \((-1, 0)\)
• Y-Intercept: \((0, 3)\)
• Vertex: \((1, 4)\)

Start your graph by plotting these points on a coordinate plane. The x-intercepts determine where the parabola crosses the x-axis, and the y-intercept shows where it crosses the y-axis. The vertex, in this case, indicates the highest point, since the parabola opens downward. Draw a smooth curve passing through these points to complete your graph.

Remember, the leading coefficient \( a = -1 \) in \( f(x) = -x^2 + 2x + 3 \) indicates the parabola opens downward. This affects the overall shape and direction of the curve, giving it a maximal vertex. Graphing parabolas helps to visualize how changes in the quadratic equation influence its graph and aids in understanding real-world scenarios represented by quadratic functions.