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Graphing quadratic equations involves plotting their functions on a coordinate plane. A quadratic equation takes the form \( ax^2 + bx + c = 0 \). When graphed, it forms a U-shaped curve known as a parabola. The way the parabola opens depends on the sign of the coefficient \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

In this case, for the equation \( y = x^2 + 2x + 5 \), we plot the equation on the graph. Choose a series of \( x \) values and compute the corresponding \( y \) values, as shown in the example provided. Plot these points and sketch a smooth curve through them to visualize the parabola.
This visualization helps us to determine the nature of the roots. If the parabola touches or crosses the x-axis, the roots are real. If it does not, the roots are complex.

The vertex of a parabola is its highest or lowest point, depending on whether it opens down or up. For a quadratic equation in standard form \( y = ax^2 + bx + c \), you can determine the vertex using the formula for the x-coordinate as \(-\frac{b}{2a}\). This gives the x position within the vertex.
For the equation \( y = x^2 + 2x + 5 \), we find the vertex by calculating:

• The x-coordinate: \(-\frac{2}{2 \times 1} = -1\)
• The y-coordinate by substituting \( x = -1 \) back into the function: \( y = (-1)^2 + 2(-1) + 5 = 4 \)

The vertex is \((-1, 4)\). Knowing this point is crucial, as it helps when sketching the parabola and understanding its general shape and position on the graph.

A quadratic equation may not always have real roots. When the parabola does not intersect with the x-axis, the equation has complex roots. Complex roots occur when the discriminant of the quadratic equation, known by the formula \( b^2 - 4ac \), is negative.
For the equation \( x^2 + 2x + 5 = 0 \), calculate the discriminant:

• Given \( a = 1 \), \( b = 2 \), and \( c = 5 \)
• Discriminant: \( 2^2 - 4 \times 1 \times 5 = 4 - 20 = -16 \)

A negative discriminant indicates non-real solutions, specifically complex roots. Complex numbers are of the form \( a + bi \), where \( i \) is the imaginary unit \( \sqrt{-1} \). Therefore, when graphing doesn’t show any x-intersections, it’s important to consider complex roots as the solution to the equation.