Problem 22 - Algebra A Combined Function

Question

Graph each quadratic function. Label the vertex and sketch and label the axis of svmmetrv. $$ f(x)=5 x^{2} $$

Step-by-Step Solution

Answer

Vertex: (0, 0); Axis of symmetry: x = 0.

Step 1: Identify the Form

The given quadratic function is in the standard form, where \( f(x) = ax^2 + bx + c \). In this case, \( a = 5 \), \( b = 0 \), and \( c = 0 \). This form is useful for identifying key features of the function.

Step 2: Determine the Vertex

For quadratic functions in the form \( ax^2 + bx + c \), the vertex can be found using the formula \( x = -\frac{b}{2a} \). Here, \( b = 0 \), so \( x = -\frac{0}{2(5)} = 0 \). Substitute \( x = 0 \) into the function to find \( f(0) = 5(0)^2 = 0 \). Thus, the vertex is at \((0, 0)\).

Step 3: Identify the Axis of Symmetry

The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is the vertical line \( x = -\frac{b}{2a} \). From Step 2, we found the axis of symmetry is \( x = 0 \).

Step 4: Sketch the Graph

To sketch the graph, start by plotting the vertex \((0, 0)\) on a coordinate plane. Since \( a = 5 \) and is positive, the parabola opens upwards. The vertex is the lowest point. Draw the axis of symmetry as a dashed vertical line at \( x = 0 \). Draw the parabola opening upwards with the vertex at the origin.

Step 5: Label the Graph

After drawing the parabola, label the vertex \((0, 0)\). Also, label the axis of symmetry as \( x = 0 \). Use arrows to indicate that the parabola continues infinitely upwards.

Key Concepts

Vertex

The vertex of a quadratic function is a critical point and can be thought of as the 'turning point' of the parabola. For the quadratic function \( f(x) = ax^2 + bx + c \), the vertex can be calculated using the formula for the x-coordinate: \( x = -\frac{b}{2a} \). This formula comes from completing the square or using calculus to find where the derivative equals zero, meaning the slope of the graph is flat at this point.
In our given function, \( f(x) = 5x^2 \), we have \( b = 0 \). This simplifies to \( x = 0 \). Plug this back into the original function to find the y-coordinate \( f(0) = 5(0)^2 = 0 \). Thus, the vertex is at \((0, 0)\).
A vertex is important as it helps in determining the parabola's minimum or maximum value. In this case, as the parabola opens upwards (since \(a = 5\) is positive), the vertex \((0, 0)\) is the lowest point on the graph. This vertex provides symmetry to the graph and helps in sketching it accurately.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror images. It passes through the vertex and can be found using the same formula as the x-coordinate of the vertex. This line is crucial as it helps in sketching and understanding the structure of the parabola. For any quadratic function \( f(x) = ax^2 + bx + c \), the equation of the axis of symmetry is \( x = -\frac{b}{2a} \). Since we are dealing with \( b = 0 \) in our function \( f(x) = 5x^2 \), the axis of symmetry is simply \( x = 0 \).
Visually, this means that if you were to fold the parabola along this line, both halves would match perfectly. The axis of symmetry is always vertical and is especially helpful in graphing because it ensures that for every point on the one side of the axis, there is a corresponding point on the other side, equidistant from the axis. When graphing the parabola, it is customary to draw this line as a dashed vertical line to emphasize its role as a mirror line for the parabola.

Standard Form

The standard form of a quadratic function is \( f(x) = ax^2 + bx + c \). This form is versatile and used for identifying various key features of the parabola such as the direction it opens, its vertex, and its axis of symmetry. In the case of \( f(x) = 5x^2 \), which is already simplified to standard form, \( a = 5 \), \( b = 0 \), and \( c = 0 \).
Here are some critical points about the standard form:

• The coefficient \( a \) dictates the direction of the parabola. If \( a \) is positive, the parabola opens upward, creating a U-shape. If \( a \) is negative, it opens downward.
• The values of \( b \) and \( c \) influence the position of the parabola on the coordinate plane but not its shape.

In this simple function, the lack of \( b \) and \( c \) terms means the parabola is centered at the origin. This simplification allows easy identification of the vertex and axis of symmetry, helping to plot the graph effortlessly.

Parabola

A parabola is the graph of a quadratic function and has a characteristic U-shape. All quadratic functions graph into parabolas on the coordinate plane. Understanding its nature is key to mastering quadratic equations.In our function \( f(x) = 5x^2 \), the graph forms a parabola that opens upwards. This direction is dictated by the positive value of \( a \).
Here are some properties of parabolas:

• The graph will always be symmetrical about the axis of symmetry.
• The vertex is the pinnacle point, and in upward opening parabolas like this, it's the lowest point.
• The distance from the vertex to the graph on either side of the axis of symmetry increases quadratically.

For this exercise, the parabola is not only simple to visualize because of the equation's simplicity, but it also clearly shows how changing \( a \) would affect the width and direction of the parabola. Larger values of \( a \) make it narrower, whereas smaller values widen it. A perfect parabola helps in identifying solutions and the nature of quadratic equations in algebraic exercises.