91Ó°ÊÓ

A parabola is a U-shaped curve that can open upwards or downwards depending on its equation. When graphing a parabola, the form of the quadratic equation is crucial. It usually looks like this: \( y = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants.

In the given equation, \( y = -x^2 + 4x \), the negative coefficient of \( x^2 \) (which is -1 here) indicates that the parabola opens downwards. To graph a parabola effectively, one needs to identify its key features, such as the vertex and the x-intercepts.

Start by plotting the vertex, which represents the highest or lowest point of the parabola. Use the x-intercepts to determine where the parabola crosses the x-axis. These intercepts are solutions to the equation when set to zero. Finally, plot a few additional points for accuracy and draw a smooth curve connecting them to complete the parabola.

Factoring quadratics is a method used to solve quadratic equations by expressing them as the product of their factors. This process can make finding the x-intercepts of a parabola straightforward.

For the quadratic equation \(-x^2 + 4x = 0\), you can factor it as \(x(-x + 4) = 0\). This involves finding two expressions that multiply together to give the original equation. When a product equals zero, it means one or both factors must also be zero.

Therefore, set each factor equal to zero: \(x = 0\) and \(-x + 4 = 0\). Solving these gives you \(x = 0\) or \(x = 4\). These are the x-values where the parabola intersects the x-axis.

The x-intercepts of a parabola are the points where the graph crosses the x-axis. This occurs when \( y = 0 \) in the equation \( y = ax^2 + bx + c \). The x-intercepts are important as they represent the solutions or roots of the quadratic equation.

When we factor \(-x^2 + 4x\) as \(x(-x + 4)\), the value of \(x\) that makes each factor zero are the x-intercepts. We found these values to be \(x = 0\) and \(x = 4\), which are points where the parabola meets the x-axis in our example.

Graphically, these intercepts help to shape our understanding of the parabola’s position and orientation on the coordinate plane.

The vertex of a parabola is the point where it either peaks or hits its lowest point, acting as the turning point of the graph. For a parabola in the form \( y = ax^2 + bx + c \), the x-coordinate of the vertex is found using the formula \( x = \frac{-b}{2a} \).

In the equation \(-x^2 + 4x\), plug in \( a = -1 \) and \( b = 4 \) into the formula: \( x = \frac{-4}{2(-1)} = 2 \). This gives us the x-coordinate of the vertex.

Substitute \( x = 2 \) back into the equation to find the y-coordinate: \( y = -2^2 + 4(2) = 4 \). Therefore, the vertex of the parabola \(-x^2 + 4x\) is at the point \((2, 4)\). This vertex helps to locate the maxima in our downward-opening parabola and is a key feature when sketching the graph.