Problem 64 Transform the function into the ... [FREE SOLUTION]

Chapter 1: Problem 64

Transform the function into the form $f(x)=c(x-h)^{2}+k,$ where $c, k,$ and $h$ are constants, by completing the square. Use graph-shifting techniques to graph the function. $$f(x)=x^{2}+2 x-2$$

Short Answer

Expert verified

The function is $f(x) = (x+1)^2 - 3$ and the graph is shifted left 1 unit and down 3 units.

Step by step solution

Identify the Quadratic Coefficients

First, identify the quadratic expression in the form of $ax^2 + bx + c$. For the given function $f(x) = x^2 + 2x - 2$, it is already in this form, with $a = 1$, $b = 2$, and $c = -2$.

Complete the Square

To transform the expression $x^2 + 2x - 2$ into the vertex form, $(x-h)^2 + k$, start by focusing on the $x^2 + 2x$ part. To "complete the square," take the coefficient of $x$, which is 2, divide it by 2 to get 1, and then square it to get $1^2 = 1$. Add and subtract this square inside the function: $f(x) = (x^2 + 2x + 1) - 1 - 2$.

Simplify the Expression

Rewrite the quadratic part as a square and simplify: $f(x) = (x+1)^2 - 3$. Now the function is in the form $c(x-h)^2 + k$ where $c = 1$, $h = -1$, and $k = -3$.

Graph-shifting Techniques

The expression $f(x) = (x + 1)^2 - 3$ indicates a parabola with vertex $(-1, -3)$. The graph should be shifted 1 unit to the left and 3 units downward from the standard parabola $y = x^2$. Since $c = 1$, the parabola opens upwards without any vertical stretch or compression.

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Key Concepts

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

Graph Shifting

Graph shifting is a powerful technique to understand how a function changes position on the coordinate plane. In algebra, particularly with quadratics, shifting allows us to visualize transformations. When we look at the function transformation given, $f(x) = (x + 1)^2 - 3$, we can see the effects of graph shifting.

The term $(x + 1)$ indicates a horizontal shift. Since it is $+1$, the plot of the parabola shifts 1 unit to the left.
The $-3$ at the end of the function causes a vertical shift, moving the whole graph down by 3 units.
These shifts help us locate the new vertex at $(-1, -3)$ from the position of a default parabola $y = x^2$ whose vertex is at the origin $(0, 0)$.

These graph transformations help in quickly sketching the graph and understanding the behavior of the function relative to the standard $y = x^2$ graph.

Quadratic Equations

Quadratic equations generally take the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. They are interesting because their graph forms a parabola, a common curve you'll often encounter. In this exercise, our focus is on converting the quadratic equation into a different form, known as the vertex form.Key characteristics of a quadratic equation are:

The midpoint of the parabola (vertex) is a crucial point of symmetry.
Solutions to the equation (if exist) are called roots or zeros of the quadratic, which are the points where the graph intersects the x-axis.
The parabola can open upwards if $a > 0$ or downwards if $a < 0$.

Understanding these basics about quadratic equations helps in solving and graphing these equations with ease and efficiency.

Parabola Vertex Form

The vertex form of a quadratic function is a useful format that reveals the vertex's position on the graph. The form is $f(x) = c(x-h)^2 + k$, where $h, k$ represent the vertex's coordinates, and $c$ affects the parabola's width and direction.Here's why it's useful:

Visible vertex: You can easily identify the vertex of the parabola, which makes sketching the graph quicker.
Simplifies graph transformations: Adjusting $h$ and $k$ allows for direct control of horizontal and vertical shifts.
Determine direction: The sign of $c$ helps you know if the parabola opens up (when $c > 0$) or down (when $c < 0$).

By completing the square, we are transforming a standard quadratic equation into this more intuitive form. For example, converting $f(x) = x^2 + 2x - 2$ into $(x + 1)^2 - 3$. This immediately tells us the vertex is at $(-1, -3)$, and the parabola opens upwards.

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91影视

Transform the function into the form \(f(x)=c(x-h)^{2}+k,\) where \(c, k,\) and \(h\) are constants, by completing the square. Use graph-shifting techniques to graph the function. $$f(x)=x^{2}+2 x-2$$

Short Answer

Step by step solution

Identify the Quadratic Coefficients

Complete the Square

Simplify the Expression

Graph-shifting Techniques

Key Concepts

Graph Shifting

Quadratic Equations

Parabola Vertex Form

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