Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 9
Express \(f(x)\) in the form \(a(x-h)^{2}+k\) $$f(x)=-3 x^{2}-6 x-5$$
Short Answer
Expert verified
The vertex form is \\(f(x) = -3(x+1)^{2} - 2\\).
Step by step solution
01
Identify the form to complete
The goal is to express the given quadratic function in the form \(a(x-h)^{2}+k\), which is known as the vertex form. To do this, we'll complete the square.
02
Factor out the leading coefficient
Factor out the leading coefficient (-3) from the x terms: \(-3(x^{2}+2x)-5\). This simplifies our equation by isolating the terms that involve x within a manipulatable structure.
03
Complete the square inside the parenthesis
Inside the parenthesis \(x^{2}+2x\), take half the coefficient of x (which is 1), square it (1), and add and subtract that square inside the parenthesis to complete the square: \(-3(x^{2}+2x+1-1)-5\).
04
Simplify the quadratic expression
Rewrite \(x^{2}+2x+1\) as \((x+1)^{2}\) using the square completed form: \(-3((x+1)^{2}-1)-5\).
05
Distribute the leading coefficient
Distribute the -3 across \((x+1)^{2}-1\): \(-3((x+1)^{2}) + 3\). This results in \(-3(x+1)^{2} + 3 - 5\).
06
Simplify constants
Combine the constant terms: \(-3(x+1)^{2} - 2\).
07
Write the expression in vertex form
The quadratic expression is now in vertex form: \(-3(x+1)^{2} - 2\). Here, \(a = -3\), \(h = -1\), and \(k = -2\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a method used in algebra to transform a quadratic equation into a form that makes it easier to analyze, particularly when looking for the vertex form. The process helps us to rewrite the equation in a way that highlights the properties of a parabola. Here's how it works, step by step:
- **Factor the coefficient of the quadratic term**: Begin by factoring out the leading coefficient from the terms involving \( x \) so that the remaining part is easier to manipulate. This ensures that the expression within the parenthesis can be completed into a perfect square.
- **Half and Square**: Take the coefficient of the linear \( x \) term (inside the parentheses), divide it by 2 and then square it. These two resulting values will help you to complete the square inside the parenthesis.
- **Add and Subtract Inside the Parenthesis**: Use the squares you've calculated by adding and subtracting them inside the parentheses. This maintains the equality of the expression but allows the creation of a perfect square trinomial.
- **Form a Perfect Square**: With the trinomial you've constructed, factor it into the perfect square binomial form \((x+1)^2\).
Quadratic Functions
Quadratic functions are polynomial functions of degree 2, generally expressed in their standard form as \( f(x) = ax^2 + bx + c \). They portray parabolic shapes on a graph, which can open upwards or downwards depending on the sign of the coefficient \( a \).The standard form is useful, but we often seek to convert it into vertex form \( a(x-h)^2+k \). This form highlights key features of the parabola:
- **Vertex**: The point \((h, k)\) which represents the peak or trough of the parabola depending on its orientation.
- **Direction**: When \( a > 0 \), the parabola opens upwards and when \( a < 0 \), it opens downwards.
- **Line of Symmetry**: The line \( x = h \), which is vertical and divides the parabola into two mirror images.
Transformations of Functions
Transformations involve altering a function to move, stretch, or shrink its graph. For quadratic functions, these transformations can be seen clearly when the expression is in vertex form.
- **Vertical Shifts**: Adjusting the value \( k \) in the vertex form causes the graph to shift up or down. If \( k \) is increased, the graph moves upward; if decreased, it shifts downward.
- **Horizontal Shifts**: Altering the value \( h \) shifts the graph left or right. A positive \( h \) moves it left, while a negative \( h \) moves it right, due to the form \((x-h)\).
- **Reflections**: If \( a \) is negative, the parabola reflects across the x-axis, flipping it upside down.
- **Vertical Stretches and Compressions**: Changing the absolute value of \( a \) affects the width of the parabola. A larger \(|a|\) compresses the parabola vertically, making it narrower, while a smaller \(|a|\) stretches it, making it wider.
