Problem 20 - Algebra A Combined Function

Question

Graph each quadratic function. Label the vertex and sketch and label the axis of symmetry. \(F(x)=(x-2)^{2}-3\)

Step-by-Step Solution

Answer

The vertex is (2, -3), and the axis of symmetry is x = 2.

Step 1: Identifying the Quadratic Form

The given quadratic function is in the vertex form: \(F(x) = (x-h)^2 + k\), where \(h\) and \(k\) are constants. For this function, \(h = 2\) and \(k = -3\).

Step 2: Finding the Vertex

For a quadratic in the form \((x-h)^2 + k\), the vertex is located at \((h, k)\). Therefore, the vertex for \(F(x) = (x-2)^2 - 3\) is at \((2, -3)\).

Step 3: Determining the Axis of Symmetry

The axis of symmetry for the quadratic equation \(F(x) = (x-h)^2 + k\) is a vertical line given by \(x = h\). For this function, the axis of symmetry is \(x = 2\).

Step 4: Choosing Points to Plot

Select points to the left and right of the vertex to get a sense of the parabola's shape. Since the vertex is at \((2, -3)\), choose values like \(x=0\), \(x=1\), \(x=3\), and \(x=4\) to substitute into the function.

Step 5: Calculating Function Values

Substitute \(x=0\) into \(F(x)\): \[ F(0) = (0-2)^2 - 3 = 4 - 3 = 1 \]Substitute \(x=1\) into \(F(x)\): \[ F(1) = (1-2)^2 - 3 = 1 - 3 = -2 \]Substitute \(x=3\) into \(F(x)\): \[ F(3) = (3-2)^2 - 3 = 1 - 3 = -2 \]Substitute \(x=4\) into \(F(x)\): \[ F(4) = (4-2)^2 - 3 = 4 - 3 = 1 \]

Step 6: Plotting the Graph

On a coordinate plane, plot the vertex \((2, -3)\) and the additional points calculated: \((0, 1)\), \((1, -2)\), \((3, -2)\), and \((4, 1)\). Draw a smooth curve through these points, shaping a parabola opening upwards.

Step 7: Labeling the Graph

Label the vertex \((2, -3)\) on the graph. Draw and label the axis of symmetry as a dashed vertical line at \(x = 2\). Make sure the parabola clearly shows symmetry across this axis.

Key Concepts

Vertex Form

The vertex form of a quadratic function provides a straightforward way to identify key features of the graph. In the vertex form, a quadratic function is expressed as \(F(x) = (x-h)^2 + k\). Here, \(h\) and \(k\) represent the vertex coordinates of the parabola. Understanding the vertex form is crucial because it allows us to quickly recognize the point where the parabola changes its direction, either reaching its highest or lowest point, depending on the orientation.

• Vertex: Located at \((h, k)\), this is the turning point of the parabola.
• Direction: The parabola opens upwards if the leading coefficient is positive and downwards if negative.
• Transformations: The values of \(h\) and \(k\) determine horizontal and vertical translations from the origin.

By identifying the vertex form, we can sketch the graph more efficiently, as it clearly outlines the vertex's placement on the coordinate plane.

Axis of Symmetry

The axis of symmetry is a critical line that runs vertically through the graph of a parabola, splitting it into two mirror image halves. This makes it easier to graph symmetrical points around the vertex. For any quadratic in vertex form \(F(x) = (x-h)^2 + k\), the axis of symmetry is given by the equation \(x = h\). This equation tells us the x-value where the parabola makes a perfect mirror split.

• Identification: Locate the \(h\) value in the vertex form equation to find the axis.
• Graphing: Draw a dashed line at \(x = h\) to visually represent the axis on the graph.
• Symmetry: Points equidistant from the axis on either side will have the same function value.

Understanding and marking the axis of symmetry helps in plot selection for further points, ensuring accuracy in shaping the parabola.

Graphing Quadratics

Graphing quadratic functions involves a series of steps to accurately depict the curve's shape and position. Starting with plotting the vertex, you use the axis of symmetry as a guide to position additional points.

• Vertex First: Start with the vertex, ensuring its position is marked as the main reference.
• Selecting Points: Choose x-values to the left and right of the vertex that will provide a clear curve shape.
• Calculating Values: Substitute selected x-values into the function to find corresponding y-values for graph plotting.
• Drawing the Parabola: Use a smooth curve to connect these points, ensuring symmetry around the axis.

Graphing allows visualization of the function’s characteristics, making it easier to understand shifts, stretches, and reflections, enhancing the learning experience for quadratic functions.

Parabola

A parabola is the U-shaped graph represented by quadratic functions. Its shape can vary depending on the coefficients in the quadratic function's formula. The vertex form, among other forms, helps to determine the specific features of a parabola.

• Standard Shape: Depicted as \(y = ax^2 + bx + c\) or similar forms.
• Vertex: The point where the parabola changes direction, easily found using the vertex form.
• Opening Direction: The parabola opens upwards for positive \(a\) and downwards for negative \(a\).
• Applications: Parabolas appear in real-world scenarios, such as projectile motion and satellite dish designs.

Learning about parabolas involves understanding these properties and utilizing them to analyze quadratic graphs effectively, shedding light on their practical applications in various fields.