Problem 35 Sketch the graph of each quadrat... [FREE SOLUTION]

Chapter 11: Problem 35

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ F(x)=2 x^{2}-5 $$

Short Answer

Expert verified

The vertex is at (0, -5) and the axis of symmetry is $ x = 0 $.

Step by step solution

Determine the Vertex

The standard form of a quadratic function is $ f(x) = ax^2 + bx + c $. In this function, $ f(x) = 2x^2 - 5 $, $ a = 2 $, $ b = 0 $, and $ c = -5 $. The formula to find the vertex of a parabola given its function is $ x = -\frac{b}{2a} $. Since $ b = 0 $, the x-coordinate of the vertex is 0. Substituting $ x = 0 $ into the function gives $ f(0) = 2 \cdot 0^2 - 5 = -5 $. Thus, the vertex is at (0, -5).

Label the Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line which passes through the vertex. For this function, since the vertex is at $ x = 0 $, the axis of symmetry is the line $ x = 0 $.

Sketch the Graph

With the vertex at (0, -5) and the axis of symmetry $ x = 0 $, you can sketch the parabola. The function $ f(x) = 2x^2 - 5 $ opens upwards because $ a = 2 > 0 $. Near the vertex, choose values such as $ x = 1 $ and $ x = -1 $, and compute $ f(1) = 2 - 5 = -3 $ and $ f(-1) = 2 - 5 = -3 $ to find additional points (1, -3) and (-1, -3). Plot these points, draw the parabola opening upwards through (0, -5), (1, -3), and (-1, -3).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex of a Parabola

In any quadratic function of the form $ f(x) = ax^2 + bx + c $, the vertex of the parabola represents the highest or lowest point of the graph, depending on whether the parabola opens upwards or downwards. It acts as a turning point from where the graph changes direction.
For the given quadratic function $ f(x) = 2x^2 - 5 $, the terms are $ a = 2 $, $ b = 0 $, and $ c = -5 $. The vertex can be calculated using the formula for its x-coordinate:

$ x = -\frac{b}{2a} $
Since $ b = 0 $, we have $ x = 0 $
The corresponding y-coordinate is found by substituting $ x = 0 $ back into the function, resulting in $ f(0) = 2 \cdot 0^2 - 5 = -5 $

Thus, the vertex of the parabola is (0, -5). This vertex signifies the minimum point for the graph since $ a = 2 $ is greater than zero, indicating the parabola opens upwards.

Axis of Symmetry

The axis of symmetry is a vertical line that runs directly through the vertex of the parabola, splitting it into two mirror-image halves. This line is crucial because it helps us understand how the parabola will look and behave, essentially acting as a mirror line.
For any quadratic function, the equation of the axis of symmetry is closely tied to the x-coordinate of the vertex. Using the same quadratic function $ f(x) = 2x^2 - 5 $, we've already determined that the vertex is located at $ x = 0 $. This tells us:

The axis of symmetry is the line $ x = 0 $.
This vertical line divides the parabola into two identical parts.
Given the symmetrical nature around this line, if you know a point on one side, you automatically know its counterpart on the other side of the axis of symmetry.

Therefore, the axis of symmetry plays a vital role in graphing since it provides a straightforward method to check the parabola's symmetry.

Parabola Graphing

Graphing a parabola involves plotting its key features 鈥� most importantly, the vertex and axis of symmetry 鈥� and understanding the direction it opens. Since the task is to sketch $ f(x) = 2x^2 - 5 $, which opens upwards, we should be aware of following steps:

Begin by plotting the vertex at (0, -5).
Draw the axis of symmetry along the line $ x = 0 $.
Use the value of $ a $, which is positive, to determine that the parabola opens upward, indicating a "smile" shape.
Select additional points near the vertex to help form the curve; for instance at $ x = 1 $ and $ x = -1 $, both providing $ f(x) = -3 $. These points are $(1, -3)$ and $(-1, -3)$.
Carefully plot these points and draw a smooth curve through them, ensuring symmetry along the axis.

By marking these elements, a clear and accurate graph of the quadratic function is realized. Recognizing these steps not only aids in plotting this specific parabola but also serves as a model for graphing any quadratic function.

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91影视

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ F(x)=2 x^{2}-5 $$

Short Answer

Step by step solution

Determine the Vertex

Label the Axis of Symmetry

Sketch the Graph

Key Concepts

Vertex of a Parabola

Axis of Symmetry

Parabola Graphing

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