Problem 22 In exercises $21-24,$ expand a... [FREE SOLUTION]

Chapter 2: Problem 22

In exercises $21-24,$ expand and graph the function. $$ g(x)--(x+3)^{2}+4 $$

Short Answer

Expert verified

The expanded function is $g(x) = x^2 + 6x + 13$. The vertex is $(-3, -4)$, and the parabola opens upwards.

Step by step solution

Understand the Function

We begin with the given function $ g(x) = (x+3)^2 + 4 $. This is a quadratic function, which means its graph will be a parabola. The expression $(x+3)^2$ indicates the function is in vertex form, where the parabola is shifted horizontally and vertically.

Expand the Quadratic Expression

To expand $(x+3)^2$, apply the formula for square of a binomial: $(a+b)^2 = a^2 + 2ab + b^2$. Here, $a = x$ and $b = 3$, so:\[(x+3)^2 = x^2 + 2 \cdot 3 \cdot x + 3^2 = x^2 + 6x + 9\]

Incorporate the Constant Term

After expanding $(x+3)^2$ to $x^2 + 6x + 9$, add the constant term from the original function, $+4$:\[g(x) = x^2 + 6x + 9 + 4\]

Simplify the Expression

Combine the like terms (the constant terms) in the expanded function:\[g(x) = x^2 + 6x + 13\]

Graph the Function

The function $ g(x) = x^2 + 6x + 13 $ is a quadratic function, meaning its graph is a parabola. Determine the vertex using the formula $x = -b/(2a)$, where $a = 1$ and $b = 6$.Calculate:\[x = -6/(2 \cdot 1) = -3\]Substitute $x = -3$ back into the equation to find $g(-3) = (-3)^2 + 6(-3) + 13 = -4$.Thus, the vertex is $(-3, -4)$, and the parabola opens upwards as the coefficient of $x^2$ is positive.

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Key Concepts

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

Vertex Form

The vertex form of a quadratic function is a compact way to express its equation, directly revealing the location of its vertex. This form is expressed as \[ f(x) = a(x-h)^2 + k \]

Here, $ (h, k) $ represents the vertex of the parabola.
The value $ a $ determines the direction and width of the parabola.

For our function $ g(x) = (x+3)^2 + 4 $, the expression identifies immediately that the vertex is at $ (-3, 4) $. Understanding that the vertex form exposes the transformation of the parabola from the origin aids in quickly graphing or analyzing the parabola's behavior. The vertex form tells us not only the vertex but also includes shifts: the horizontal shift opposite the sign in $ (x+3) $, and the vertical shift up by 4 units.

Expanding Binomials

Expanding binomials is crucial in transforming a quadratic function from vertex form to standard form, making it easier to analyze and graph. To expand $ (x+3)^2 $, use the formula:\[ (a+b)^2 = a^2 + 2ab + b^2 \]

Set $ a = x $ and $ b = 3 $.
Applying the formula results in $ x^2 + 2 \times 3 \times x + 3^2 $.
This simplifies to $ x^2 + 6x + 9 $.

Expanding in this manner is a skill frequently used to convert vertex form equations into standard form equations. It helps us see the effect of each component鈥攍inear and constant terms separately.

Graphing Parabolas

Graphing parabolas involves plotting a smooth curve that fits the standard form of a quadratic equation. Given the equation $ g(x) = x^2 + 6x + 13 $, we know:

The coefficient of $ x^2 $ is positive, so the parabola opens upwards.
Using the result from expanding, we identify the key features of the graph.
We know the vertex, $ (-3, -4) $, is essential for plotting.

Once the vertex is plotted, we check the y-intercept by setting $ x = 0 $, yielding $ g(0) = 13 $. This helps complete the graph, ensuring that the parabola is accurately represented on the coordinate plane. These elements, combined with calculated points, help sketch a precise graph.

Vertex Calculation

Calculating the vertex of a parabola in standard form involves using the formula $ x = -b/(2a) $. For our expanded function $ g(x) = x^2 + 6x + 13 $, this calculation proves its vertex:

Identify $ a = 1 $ and $ b = 6 $ from the function.
Substitute into the formula: $ x = -6/(2 \times 1) = -3 $.
This calculation confirms the horizontal location of the vertex.

Next, substitute $ x = -3 $ back into the function to find $ g(-3) $, showing $ (-3)^2 + 6(-3) + 13 = -4 $. Thus, the vertex is indeed $ (-3, -4) $. Knowing how to calculate this precisely allows accurate graphing and understanding of the parabola's shape and position.

91影视

In exercises \(21-24,\) expand and graph the function. $$ g(x)--(x+3)^{2}+4 $$

Short Answer

Step by step solution

Understand the Function

Expand the Quadratic Expression

Incorporate the Constant Term

Simplify the Expression

Graph the Function

Key Concepts

Vertex Form

Expanding Binomials

Graphing Parabolas

Vertex Calculation

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