Problem 25 - Algebra A Combined Function

Question

Graph each quadratic function. Label the vertex and sketch and label the axis of svmmetrv. $$ H(x)=2 x^{2} $$

Step-by-Step Solution

Answer

The vertex is at (0,0), the axis of symmetry is x=0, and the parabola opens upwards.

Step 1: Identify the Coefficients

The quadratic function given is written in the form \( H(x) = ax^2 + bx + c \). Here, \( a = 2 \), \( b = 0 \), and \( c = 0 \).

Step 2: Find the Vertex

The vertex of the parabola given by \( H(x) = ax^2 + bx + c \) can be calculated using the formula \( x = -\frac{b}{2a} \). For our function, \( b = 0 \) and \( a = 2 \), so \( x = -\frac{0}{4} = 0 \). Substitute back into \( H(x) \) to get \( H(0) = 2(0)^2 = 0 \). Therefore, the vertex is at the point \( (0,0) \).

Step 3: Determine the Axis of Symmetry

The axis of symmetry for a quadratic function is always a vertical line passing through the x-value of the vertex. Here, since the vertex is at \( (0,0) \), the axis of symmetry is the line \( x = 0 \).

Step 4: Sketch the Graph

This function represents a parabola that opens upwards because \( a = 2 > 0 \). Plot the vertex at \( (0,0) \), and choose points on either side of the vertex to determine the shape of the parabola. For example, at \( x = 1 \), \( H(1) = 2(1)^2 = 2 \), and at \( x = -1 \), \( H(-1) = 2(-1)^2 = 2 \). Plot these points and draw the parabola, making sure it's symmetric about the axis \( x = 0 \).

Step 5: Label the Graph

Label the vertex on the graph as \( (0,0) \) and the axis of symmetry as the line \( x=0 \).

Key Concepts

Vertex Calculation

In the context of quadratic functions, the vertex is a critical point where the graph changes direction. Calculating the vertex provides important insights for graphing the parabola. The given quadratic function is in the form \( H(x) = ax^2 + bx + c \). The vertex \( x \)-coordinate is found using the formula:

• \( x = -\frac{b}{2a} \)

For the function \( H(x) = 2x^2 \), \( a = 2 \) and \( b = 0 \). Plug these values into the formula to get:

• \( x = -\frac{0}{4} = 0 \)

To get the \( y \)-coordinate, substitute \( x = 0 \) back into the equation:

• \( H(0) = 2(0)^2 = 0 \)

Thus, the vertex is at \((0, 0)\). This point is vital as it shows where the parabola begins its characteristic curve.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that divides it into two mirror-image halves. Understanding this axis helps in making an accurate sketch of the quadratic graph.The line of symmetry can be found using the vertex \( x \)-coordinate. For a quadratic equation \( ax^2 + bx + c \), the axis of symmetry is given by:

• \( x = -\frac{b}{2a} \)

Since the vertex for our function \( H(x) = 2x^2 \) is at \((0, 0)\), the axis of symmetry is:

• \( x = 0 \)

This line tells us that the parabola is perfectly symmetric across the y-axis, and helps determine that each point \( (x, y) \) on one side has a corresponding point \((-x, y)\) on the other.

Parabola

A parabola is the graph of a quadratic function and is known for its unique U-shape. In this exercise, the function \( H(x) = 2x^2 \) is a classic example of such a curve.Here are some key features of parabolas:

• The vertex is the tip of the parabola, which we calculated to be at \((0, 0)\).
• Parabolas can open upwards or downwards.

For the function \( H(x) = 2x^2 \), the parabola opens upwards because its leading coefficient is positive \( (a = 2) \). Graphically, this means it curves away from the vertex as \( x \) moves in both directions from the center (i.e., the axis of symmetry).Understanding these characteristics helps in sketching and predicting the behavior of quadratic graphs.

Quadratic Coefficients

The coefficients in a quadratic function determine the shape and position of the parabola. For the function \( H(x) = ax^2 + bx + c \), the values of \( a \), \( b \), and \( c \) each play roles:

• \( a \): Determines the direction and width of the parabola. If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards. Larger \( |a| \) values mean a narrower parabola.
• \( b \): Affects the position of the vertex along the x-axis.
• \( c \): Represents the y-intercept, the point where the parabola crosses the y-axis.

In \( H(x) = 2x^2 \):

• \( a = 2 \): Since \( a \) is positive, the parabola opens upwards and is relatively narrow.
• \( b = 0 \) and \( c = 0 \): Together, they make the vertex lie at the origin \((0, 0)\), giving a symmetrical graph.

Upward Opening Parabola

An upward opening parabola is one of two possible orientations of a parabola, characterized by its open end facing upwards. The function \( H(x) = 2x^2 \) serves as a perfect example of this type.The essential properties of an upward opening parabola include:

• The value of \( a > 0 \) determines this orientation, making the arms of the parabola rise as you move away from the vertex.
• The vertex is a minimum point on the graph, indicating the lowest y-value on the parabola.

In our example, the upward opening nature of \( H(x) = 2x^2 \) indicates that the lowest point is the vertex at \((0,0)\), and the graph extends infinitely upwards as \( x \) moves both positively and negatively.Recognizing this pattern helps in assessing and sketching quadratic graphs accurately.