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Magazine Chapter 2: Problem 10
Express \(f(x)\) in the form \(a(x-h)^{2}+k\) $$f(x)=-4 x^{2}+16 x-13$$
Short Answer
Expert verified
\(f(x) = -4(x - 2)^2 + 3\).
Step by step solution
01
Expand the Quadratic Formula
The given function is \(f(x) = -4x^2 + 16x - 13\). We want to express it in the form \(a(x-h)^2 + k\), which is a vertex form of a parabola.
02
Factor Out the Coefficient of \(x^2\)
Begin by factoring the coefficient of \(x^2\), which is -4, out of the first two terms: \(-4(x^2 - 4x) - 13\).
03
Complete the Square
To complete the square inside the parenthesis, take half of the coefficient of \(x\) from \(x^2 - 4x\), which is -2, and square it to get 4. Add and subtract this square inside the parenthesis: \(-4(x^2 - 4x + 4 - 4) - 13\).
04
Simplify the Expression
Rewrite the expression inside the parenthesis as a perfect square and simplify: \(-4((x - 2)^2 - 4) - 13\).
05
Distribute and Combine Constants
Distribute the -4: \(-4(x - 2)^2 + 16 - 13\). Simplify by combining the constants: \(-4(x - 2)^2 + 3\).
06
Final Vertex Form
The function in vertex form is \(f(x) = -4(x - 2)^2 + 3\), where \(a = -4\), \(h = 2\), and \(k = 3\).
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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 handy algebraic technique for transforming a quadratic equation into its vertex form. This method is especially useful for converting the general quadratic equation into a form that reveals important features of the graph such as the vertex, axis of symmetry, and direction of opening.
To complete the square, follow these steps:
To complete the square, follow these steps:
- Start with a quadratic expression in the form of \( ax^2 + bx + c \).
- If \(aeq1\), factor \(a\) out from the first two terms, so it looks like \( a(x^2 + \frac{b}{a}x) + c \).
- Find half of the coefficient of \(x\) (i.e., \(\frac{b}{2a}\)), then square it and add and subtract this value inside the parenthesis.
- Rearrange the expression inside the bracket as a squared binomial.
- Finally, multiply through by the factored term and simplify any constants outside the brackets.
Quadratic Functions
Quadratic functions represent polynomial equations of the form \( ax^2 + bx + c \), where \(aeq0\). They are fundamental in algebra and appear frequently in various fields like physics, engineering, and economics.
Key characteristics of quadratic functions include:
Key characteristics of quadratic functions include:
- The graph of a quadratic function is a parabola.
- The leading coefficient \(a\) determines the direction of the parabola's opening: upward if \(a>0\) and downward if \(a<0\).
- The vertex of the parabola is an essential feature as it denotes the highest or lowest point of the curve.
- The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), can solve for the roots of the function.
Parabola
A parabola is the geometric representation of a quadratic function. It is a symmetric curve that can open either upwards or downwards depending on the sign of the leading coefficient. Each parabola has key elements and characteristics worth noting.
Important properties of a parabola include:
Important properties of a parabola include:
- The vertex, which signifies the peak or trough of the parabola, derived easily from the vertex form of a quadratic \( a(x-h)^2+k \).
- The axis of symmetry is a vertical line passing through the vertex, given by \(x = h\).
- The direction of opening: If the coefficient \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.
- It can cross the x-axis at one or two points, known as the roots or zeros.
