Problem 6 - 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. $$ y=-2 x^{2} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Coefficients

The given equation is in the standard form of a parabola, which is \(y = ax^2 + bx + c\). For the equation \(y = -2x^2\), identify the coefficients: \(a = -2\), \(b = 0\), and \(c = 0\).

Step 2: Find the Vertex Formula

The vertex \((h, k)\) of a parabola in the form \(y = ax^2 + bx + c\) can be found using the formula \(h = -\frac{b}{2a}\).

Step 3: Calculate the Vertex's x-coordinate

Substitute the values of \(a\) and \(b\) into the formula for \(h\):\[h = -\frac{0}{2(-2)} = 0\]

Step 4: Calculate the Vertex's y-coordinate

Substitute the x-coordinate back into the original equation \(y = -2x^2\) to find \(k\):\[k = -2(0)^2 = 0\]

Step 5: State the Vertex

The vertex of the parabola is \((0, 0)\).

Step 6: Graph the Parabola

Since the equation is \(y = -2x^2\), it represents a parabola that opens downward (because \(a < 0\)) with the vertex at \((0, 0)\). Draw the axis of symmetry at \(x = 0\) and sketch the parabola opening downward.

Key Concepts

Parabola Graphing

Graphing a parabola involves plotting points that satisfy the equation, particularly focusing on the vertex and curvature. For any parabolic equation like \(y = -2x^2\), the parabola is shaped like a U (or an upside-down U if the coefficient \(a\) is negative). This upside-down shape is typical because of the negative coefficient \(a = -2\).
To graph this parabola, we need to consider the vertex, which is the highest or lowest point on the parabola, depending on its orientation. Begin by marking the vertex on a graph, here at (0, 0).

• Draw the axis of symmetry: For our parabola, this vertical line runs along \(x = 0\).
• Since \(a = -2\), the graph opens downward, showing a more intense inward curve opposite to where the parabola opens when \(a\) is positive.
• Additional points can be chosen on either side of the axis (e.g., \(x = 1, -1\)) to confirm the precise shape by plugging them back into the equation.

Once these points are plotted, simply connect them with a smooth curve, ensuring the graph accurately represents the parabola's symmetrical nature and its concave shape due to the negative \(a\) value.

Quadratic Functions

Quadratic functions are mathematical expressions of the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These functions create a parabolic graph. Understanding their structure helps in predicting and analyzing graphed outcomes.
In a quadratic function:

• \(a\) determines the direction of the parabola's opening. If \(a > 0\), it opens upwards. If \(a < 0\), it opens downwards.
• \(b\) influences the position of the vertex along the horizontal axis, but in \(y = -2x^2\), \(b = 0\), meaning the graph is symmetric around the y-axis.
• \(c\) provides the y-intercept, the point where the parabola crosses the y-axis. For our example, \(c = 0\), which means it passes through the origin.

These elements \(a\), \(b\), and \(c\) enable calculations like locating the vertex or determining the y-intercept directly from the equation, making it easier to predict the parabola's behavior on the graph. Understanding these fundamentals allows for manipulating the equation for specific graph interpretations or alterations.

Vertex Form of a Parabola

The vertex form of a parabola introduces a clear way to find its vertex on a graph. This form appears as \(y = a(x-h)^2 + k\), where \((h, k)\) represents the vertex. Converting from standard form \(y = ax^2 + bx + c\) to vertex form, helps in instantly identifying the key components.

• The vertex \((h, k)\) in vertex form relates directly to the maximum or minimum point of a parabola.
• For standard form parabolas like \(y = -2x^2\) or more complex ones, you can extract the vertex using \(h = -\frac{b}{2a}\) and then substitute back to find \(k\).

Rewriting \(y = -2x^2\) into vertex form might read like \(y = -2(x-0)^2 + 0\), reflecting its vertex at \((0,0)\). This simplicity assures easier graphing and analysis, especially when needing to make shifts horizontally or vertically or stretch the curve, all evident when using vertex form directly.