Problem 2 For each of the quadratic functi... [FREE SOLUTION]

Chapter 3: Problem 2

For each of the quadratic functions fin questions $1-5,$ find the following: a) the axis of symmetry and the vertex, by algebraic methods b) the transformation(s) that can be applied to $y=x^{2}$ to obtain the graph of $y=f(x)$ c) the minimum or maximum value of $f$ Check your results using your GDC $$f: x \mapsto x^{2}+6 x+8$$

Short Answer

Expert verified

Axis of symmetry: $ x = -3 $, Vertex: $(-3, -1)$, Minimum value: $-1$.

Step by step solution

Identify the coefficients

The given quadratic function is in the form of $ f(x) = ax^2 + bx + c $. Here, $ a = 1 $, $ b = 6 $, and $ c = 8 $. This is important for finding the axis of symmetry and vertex.

Find the axis of symmetry

The axis of symmetry for a quadratic function $ f(x) = ax^2 + bx + c $ is given by the formula $ x = -\frac{b}{2a} $. Substituting $ b = 6 $ and $ a = 1 $ into the formula, we get: \[ x = -\frac{6}{2(1)} = -3 \]. So, the axis of symmetry is $ x = -3 $.

Find the vertex

Substitute $ x = -3 $ into the original function to find the $ y $-coordinate of the vertex. $ f(-3) = (-3)^2 + 6(-3) + 8 = 9 - 18 + 8 = -1 $. Therefore, the vertex is $ (-3, -1) $.

Describe the transformations

Start with the parent function $ y = x^2 $. The function $ f(x) = x^2 + 6x + 8 $ can be rewritten by completing the square or seen from the vertex form as a shift: It is a translation left by 3 units and down by 1 unit. Therefore, the transformations are translations left 3 units and down 1 unit.

Determine the minimum/maximum value of f

Since the coefficient $ a = 1 $ is positive, the parabola opens upward and the vertex represents the minimum point. Thus, the minimum value of $ f(x) $ is $-1$, which occurs at $ x = -3 $.

Verify results using GDC

Input the function $ y = x^2 + 6x + 8 $ into a Graphical Display Calculator (GDC) to visually confirm the graph's vertex and axis of symmetry. The graph should show the vertex at $(-3, -1)$ and confirm the axis of symmetry at $ x = -3 $.

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Key Concepts

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

Axis of Symmetry

The axis of symmetry is an important feature of quadratic functions. It is a vertical line that divides the parabola into two mirror images. The equation for finding the axis of symmetry in a quadratic function, written as $ f(x) = ax^2 + bx + c $, is given by:

$ x = -\frac{b}{2a} $

In the given exercise, we identified the coefficients as $ a = 1 $, $ b = 6 $, and $ c = 8 $. By substituting $ b = 6 $ and $ a = 1 $ into the formula, we find the axis of symmetry:

$ x = -\frac{6}{2(1)} = -3 $

This means the line of symmetry is at $ x = -3 $, splitting the parabola into two equal halves. Understanding this concept helps in analyzing the graph's balance and predicting its behavior.

Vertex

The vertex of a parabola is its highest or lowest point, depending on the direction it opens. For a quadratic function, it can be found using the axis of symmetry. Once the $ x $-coordinate of the vertex is found, we substitute it back into the function to find the $ y $-coordinate. In this exercise, we found:

$ x = -3 $ from the axis of symmetry
Substituting $ x = -3 $ into $ f(x) $ gives: $ f(-3) = (-3)^2 + 6(-3) + 8 = -1 $
The vertex is $(-3, -1)$

This point provides the location where the parabola changes direction, also known as the turning point. Identifying the vertex is crucial for understanding the overall shape and position of the parabola.

Parabola Transformations

Parabola transformations involve changes in the position and orientation of the graph from its simplest form, $ y = x^2 $. The transformations applied in the provided function are a result of its equation $ f(x) = x^2 + 6x + 8 $. Let's consider these transformations:

Start with the basic function $ y = x^2 $
Due to the term $ 6x $, there's a horizontal shift (left 3 units), as identified by the vertex $ x = -3 $
The constant 8 in the function influences vertical shifts, resulting in a downward shift by 1 unit, evident from the vertex $ y = -1 $

These shifts are translations in terms of transformation. Recognizing these allows us to plot the graph without the need for computation and visualize the parabolic shift easily.

Minimum or Maximum Value

The parabolic graph of a quadratic function will have either a minimum or maximum value. This depends on the sign of the leading coefficient $ a $. If $ a > 0 $, the parabola opens upwards, indicating a minimum value at its vertex. If $ a < 0 $, the parabola opens downwards, showing a maximum value. Given $ f(x) = x^2 + 6x + 8 $, where $ a = 1 $ (positive), the function opens upwards.

The vertex $(-3, -1)$ provides the lowest point on the graph, hence it is the minimum value.
This minimum value of $ f(x) = -1 $ occurs when $ x = -3 $

Understanding this concept helps determine important graph characteristics critical in applications like optimization and modeling.

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Short Answer

Step by step solution

Identify the coefficients

Find the axis of symmetry

Find the vertex

Describe the transformations

Determine the minimum/maximum value of f

Verify results using GDC

Key Concepts

Axis of Symmetry

Vertex

Parabola Transformations

Minimum or Maximum Value

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