Problem 5 - Algebra A Combined Function

Question

The graph of each equation is a parabola. Find the vertex of the parabola and then graph it. See Examples 1 through 4. $$ y=-4 x^{2} $$

Step-by-Step Solution

Answer

The vertex of the parabola is (0, 0), and it opens downward.

Step 1: Identify the Parabola Form

The given equation is in the form of \( y = ax^2 + bx + c \), where \( a = -4 \), \( b = 0 \), and \( c = 0 \). This is a standard form of a quadratic equation representing a parabola.

Step 2: Determine the Vertex Formula

In a parabola given by \( y = ax^2 + bx + c \), the x-coordinate of the vertex can be found using \( x = -\frac{b}{2a} \). For our equation, \( b = 0 \), so the formula simplifies to \( x = 0 \).

Step 3: Calculate the Vertex

Substitute \( x = 0 \) back into the equation \( y = -4x^2 \) to find the y-coordinate of the vertex. We get: \( y = -4 \times 0^2 = 0 \). Thus, the vertex is \( (0, 0) \).

Step 4: Graph the Parabola

Since the vertex is at \( (0, 0) \) and the parabola opens downwards (because \( a = -4 \) is negative), plot the vertex on the graph. Points on either side can be chosen, such as \( x = 1 \) and \( x = -1 \), to find \( y \) values: \( y = -4 \times 1^2 = -4 \) and \( y = -4 \times (-1)^2 = -4 \), yielding points \((1, -4)\) and \((-1, -4)\). Plot these points and draw the parabola opening downwards.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that has the highest degree of 2, represented by the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This means the equation includes a term with \( x^2 \), and it can represent a variety of curves when graphed. Quadratic equations are fundamental because they form parabolas which can open upwards or downwards depending on the sign of \( a \). If \( a \) is positive, the parabola opens upwards forming a U-shape; if \( a \) is negative, it opens downwards, resembling an inverted U.

Standard Form of a Quadratic

The standard form of a quadratic equation is a specific way to express these equations as \( y = ax^2 + bx + c \). This form helps in identifying key features of the parabola, such as direction and the vertex, more easily. For the exercise provided, \( y = -4x^2 \), the equation is already in standard form, where \( a = -4 \), \( b = 0 \), and \( c = 0 \). Remember: in the standard form, \( a \) indicates the "width" and direction of the parabola. A larger absolute value of \( a \) denotes a narrower graph, while a smaller absolute value means a wider one.

Graphing a Parabola

Graphing a parabola involves plotting points derived from the quadratic equation and sketching the curve formed by these points. First, you identify the vertex—the turning point of the parabola. This is crucial as it helps structure the graph. For \( y = -4x^2 \), we have determined that the vertex is at \((0,0)\), meaning it is at the origin on the graph. Since the parabola opens downwards due to \( a = -4 \), we know it takes the shape of an upside-down U.

• Identify the vertex
• Determine the direction by the sign of \( a \)
• Choose points around the vertex (e.g., \( x = 1 \) and \( x = -1 \)) to get symmetric points for plotting

By following these steps, you can accurately graph the parabola.

Parabola Vertex Formula

The vertex of a parabola is a key point and is found using the vertex formula. In the equation, \( y = ax^2 + bx + c \), the vertex’s x-coordinate is given by \( x = -\frac{b}{2a} \). Once you find \( x \), you substitute it back into the original equation to find \( y \). This gives you the coordinates \((x, y)\) of the vertex. In our example, with \( b = 0 \) and \( a = -4 \), the x-coordinate simplifies to \( x = 0 \). Plugging this back into the equation \( y = -4x^2 \), we confirm that \( y = 0 \). Therefore, the vertex is at \((0, 0)\). This is the point where the parabola changes direction from decreasing to increasing or vice versa.