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Quadratic equations are a vital aspect of algebra that deals with equations in the form of \( y = ax^2 + bx + c \). This type of equation is known as a polynomial equation of degree two, implying that the highest power of \( x \) is 2. In the context of graphing, this type of equation creates a curve known as a parabola. The way this parabola opens and its particular shape depends on the values of \( a \), \( b \), and \( c \).

• If \( a \) is positive, the parabola opens upwards, resembling a U-shape.
• If \( a \) is negative, it opens downwards, resembling an upside-down U-shape.

Understanding these characteristics helps in plotting the quadratic functions accurately on a graph and interpreting their physical representation.

The vertex of a parabola is a crucial point that represents the peak or the lowest point of the curve, depending on whether it opens upwards or downwards. To find the vertex, you can use the formula \[x = -\frac{b}{2a}\] which gives the x-coordinate of the vertex. In our specific equation \( y = -x^2 + 4x - 5 \), substituting \( a = -1 \) and \( b = 4 \) yields \[x = -\frac{4}{-2} = 2\].
Once you have the x-coordinate, plug it back into the original quadratic equation to find the y-coordinate. For this equation:

• Substitute \( x = 2 \) back into the equation: \( y = -(2)^2 + 4(2) - 5 = -1 \)
• The vertex, therefore, is \( (2, -1) \).

Understanding the vertex is vital because it gives critical information about the overarching shape and maximum or minimum points of the parabola.

The axis of symmetry is a line that divides a parabola into two mirrored halves. It is a critical property when graphing any quadratic equation. For a parabola given by the quadratic equation \( y = ax^2 + bx + c \), the axis of symmetry can be found using the same calculation as the x-coordinate of the vertex:\[x = -\frac{b}{2a}\]This vertical line runs through the vertex, ensuring that each point on one side of the parabola has a matching point directly opposite on the other side.
In the equation \( y = -x^2 + 4x - 5 \), using \( a = -1 \) and \( b = 4 \), the axis of symmetry is:

• \( x = 2 \)

Knowing this line helps you accurately graph the parabola and provides insight into its shape and direction.

Intercepts are points where the graph intersects the axes. These include the y-intercept and the potential x-intercepts.For the **y-intercept**:- It occurs where the graph crosses the y-axis (i.e., where \( x=0 \)).- Substitute \( x = 0 \) into the equation: \( y = -(0)^2 + 4(0) - 5 = -5 \).- So, the y-intercept for this equation is \( (0, -5) \).The **x-intercepts** are points where the graph crosses the x-axis (i.e., where \( y=0 \)). However, in this quadratic equation:\[ 0 = -x^2 + 4x - 5 \]Attempting to calculate the x-intercepts using the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]leads to a discriminant of \( 16 - 20 = -4 \), indicating that there are no real x-intercepts for this equation.Understanding intercepts aids in the graphing process and gives additional insights into the locations where the function interacts with the axes.