/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 The vertex of \(y=a x^{2}+b x+c\... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 1

The vertex of \(y=a x^{2}+b x+c\) is at \(x=-b / 2 a\). What is special about this \(x\) ? Show that it gives \(y=c-\left(b^{2} / 4 a\right)\).

Short Answer

Expert verified
The vertex \( x \) gives the minimum/maximum \( y = c - \frac{b^2}{4a} \) of the parabola.

Step by step solution

01

Determine the Expression for Vertex

The vertex of a parabola given by the function \( y = ax^2 + bx + c \) can be determined by the formula \( x = \frac{-b}{2a} \). This \( x \) value represents the point where the parabola reaches its minimum (if \( a > 0 \)) or maximum (if \( a < 0 \)). It is the point of symmetry for the graph of the parabola.
02

Substitute the x-value into the Original Equation

Substitute \( x = \frac{-b}{2a} \) into the quadratic function \( y = ax^2 + bx + c \) to find the corresponding \( y \)-value. Replace every instance of \( x \) with \( \frac{-b}{2a} \).
03

Simplify the Expression

Substitute \( x = \frac{-b}{2a} \) in \( y = ax^2 + bx + c \) and simplify:\[ y = a \left( \frac{-b}{2a} \right)^2 + b \left( \frac{-b}{2a} \right) + c \]Simplify inside the parentheses:\[ y = a \left( \frac{b^2}{4a^2} \right) - \frac{b^2}{2a} + c \]Combine the terms:\[ y = \frac{ab^2}{4a^2} - \frac{b^2}{2a} + c \]Which simplifies to:\[ y = \frac{b^2}{4a} - \frac{2b^2}{4a} + c \]\[ y = c - \frac{b^2}{4a} \].
04

Confirm the Simplification

After simplifying, we have shown that at the vertex, the \( y \)-value is \( y = c - \frac{b^2}{4a} \). This result confirms that the value of \( y \) at \( x = \frac{-b}{2a} \) is special because it reflects the minimum or maximum value of the parabola, depending on the sign of \( a \).

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Key Concepts

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

Quadratic Function
A quadratic function is a type of polynomial function that is characterized by its highest degree being 2. It is generally expressed in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This equation forms the graph of a parabola when plotted on a coordinate plane.
  • \( a \): Determines the width and direction of the parabola. If it's positive, the parabola opens upwards, indicating a minimum point. If negative, it opens downwards, indicating a maximum point.
  • \( b \): Affects the position of the vertex horizontally. It impacts the symmetry axis of the parabola.
  • \( c \): Represents the y-intercept of the parabola. It's where the graph intersects the y-axis.
Understanding these components helps in graphing the parabola and predicting its shape and orientation. When solving for different features such as the vertex or intercepts, manipulating these constants in the quadratic equation is crucial.
Symmetry of a Parabola
The symmetry of a parabola is a fascinating aspect that makes it a neat geometric figure. A parabola is symmetric concerning its vertex, meaning you can draw a line through the vertex called the axis of symmetry, and both halves of the parabola will mirror each other. The equation of this line is given by \( x = \frac{-b}{2a} \). Every parabola has exactly one axis of symmetry, and understanding this helps in predicting the parabola's precise shape and plotting it more efficiently. Notably, locating the axis of symmetry:
  • Helps in determining the vertex, as it passes through the vertex point.
  • Ensures that the values on one side of the parabola are mirrored on the other side.
  • Assists in graph balancing, especially when calculating symmetric points for plots.
The concept of symmetry in a parabola is pivotal in various mathematical applications, including optimization problems and physics-related trajectory analyses.
Maximum and Minimum of a Parabola
Understanding the maximum and minimum values of a parabola is essential when analyzing a quadratic function. These points indicate the highest or lowest point on the curve, which can be of great significance depending on the application.
  • For a parabola that opens upwards \((a > 0)\), the vertex represents the minimum point on the parabola. Thus, the y-coordinate of the vertex is the smallest value that the function can attain.
  • Conversely, for a parabola that opens downwards \((a < 0)\), the vertex signifies the maximum point, where the y-coordinate is the largest value.
The vertex's y-coordinate value is given by \( y = c - \frac{b^2}{4a} \), as shown in the simplification steps. This calculation allows us to predict these extremum points accurately. These maximum and minimum values are crucial in areas such as optimization, economics, and any field requiring identification of peak or trough values in a given scenario.

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