Problem 44 Complete the square to write eac... [FREE SOLUTION]

Chapter 10: Problem 44

Complete the square to write each function in $f(x)=a(x-h)^{2}+k$ form. Determine the vertex and the axis of symmetry of the graph of the function. Then plot several points and complete the graph. See Examples 6 and 7 . $$ f(x)=x^{2}+6 x+5 $$

Short Answer

Expert verified

The function in vertex form is $ f(x) = (x+3)^2 - 4 $, with vertex $(-3, -4)$ and axis of symmetry $ x = -3 $.

Step by step solution

Rearrange the function

The given function is $ f(x) = x^2 + 6x + 5 $, which includes a quadratic term. Since there is no coefficient other than 1 in front of $ x^2 $, we can proceed directly into completing the square without needing to factor out a term.

Complete the square

To complete the square, take the coefficient of the $ x $ term, which is 6, halve it to get 3, and then square it to get 9. Add and subtract 9 inside the function:\[ f(x) = x^2 + 6x + 9 - 9 + 5 \] Now, isolate the perfect square trinomial:

Write as a perfect square

The expression $ x^2 + 6x + 9 $ is a perfect square that can be written as $(x+3)^2$. Incorporate this back into the expression:\[ f(x) = (x+3)^2 - 9 + 5 \] Simplify the constants:\[ f(x) = (x+3)^2 - 4 \]

Identify vertex and axis of symmetry

The function is now in the form $ f(x) = a(x-h)^2 + k $, where $ a = 1 $, $ h = -3 $, and $ k = -4 $. Thus, the vertex is $(-3, -4)$. The axis of symmetry is the vertical line $ x = -3 $.

Graph the function

On a graph, the vertex $(-3, -4)$ is the lowest point since $ a = 1 > 0 $, indicating a parabola opening upwards. Plot the vertex, and find additional points by choosing x-values around the vertex. For example:- $ f(-4) = (-4 + 3)^2 - 4 = 1 - 4 = -3 $- $ f(-2) = (-2 + 3)^2 - 4 = 1 - 4 = -3 $Plot these points and draw the parabola with symmetry around $ x = -3 $.

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Key Concepts

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

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a specific form that reveals key properties of a parabola. This method involves creating a perfect square trinomial from a quadratic expression. Let鈥檚 see how this works step by step for the expression given in the exercise.
First, consider the quadratic function:

Take the middle coefficient from the term with the variable x, which in this case is 6 for the function $x^2 + 6x + 5$.
Halve it, resulting in 3, then square it to get 9.
Now, add and subtract this square (9) within the function. This gives us: \[ f(x) = x^2 + 6x + 9 - 9 + 5 \]

This step of adding and subtracting the same number allows us to transform part of the expression into a perfect square trinomial. We are left with:

$ (x+3)^2 - 4 $

This step rearranges the expression nicely, making it much easier to analyze and graph later.

Vertex Form

The vertex form of a quadratic function is very useful for identifying the key features of the parabola it represents. This form is given by: \[ f(x) = a(x-h)^2 + k \] where

$a$ is a nonzero constant,
$ (h, k) $ is the vertex of the parabola.

From the completed square in our function,
$ f(x) = (x+3)^2 - 4 $, we can see that:

$h = -3$
$k = -4$,

Thus, the vertex is $(-3, -4)$. The form $ a(x-h)^2 + k $ represents how the graph shifts.

It's translated left by 3 units and down by 4 units.

This transformation makes it extremely simple to sketch the graph and pinpoint its vertex. The vertex indicates the turning point of the parabola and tells us where the parabola changes direction.

Parabola Graphing

Graphing a parabola involves plotting its main points and understanding its shape. Begin with the vertex, the most critical point in the vertex form, here found to be $(-3, -4)$. This point acts as the center of the graph. Since the coefficient $a = 1 > 0$, the parabola opens upwards.
To plot the parabola:

First, plot the vertex $(-3, -4)$ on a coordinate graph.
Take some additional x-values around the vertex to find more points. For example, choose $ x = -4 $ and $ x = -2 $.
Calculate the corresponding y-values using the function:
$ f(-4) = 1 - 4 = -3 $
$ f(-2) = 1 - 4 = -3 $
Plot these points $(-4, -3)$, $(-2, -3)$ on the graph.

With these points, you can draw a smooth curve connecting them. The parabola will have a symmetric and smooth U-shaped curve centered around the vertex, matching the axis of symmetry.

Axis of Symmetry

The axis of symmetry is a vital part of understanding parabolic graphs. It is a vertical line that divides the parabola into two mirror-image halves. For a quadratic function in vertex form $f(x) = a(x-h)^2 + k$, the axis of symmetry is the line $x = h$.
In our exercise, with the vertex found at $(-3, -4)$, the axis of symmetry is the line $x = -3$.

This means the parabola is symmetric around this vertical line.
Each point on the left of this line has a corresponding point mirrored to the right.
For our parabola, $ (-4, -3) $ and $-2, -3)$ are such mirrored points based on the calculations.

This symmetry is crucial for graphing because it assures that once part of the parabola is plotted, the other side naturally follows.

It also helps determine additional points when constructing the graph.

91影视

Complete the square to write each function in \(f(x)=a(x-h)^{2}+k\) form. Determine the vertex and the axis of symmetry of the graph of the function. Then plot several points and complete the graph. See Examples 6 and 7 . $$ f(x)=x^{2}+6 x+5 $$

Short Answer

Step by step solution

Rearrange the function

Complete the square

Write as a perfect square

Identify vertex and axis of symmetry

Graph the function

Key Concepts

Completing the Square

Vertex Form

Parabola Graphing

Axis of Symmetry

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