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Chapter 3: Problem 19

Write each quadratic function in the form \(y=a(x-h)^{2}+k\) and sketch its graph. $$y=-2 x^{2}+3 x-1$$

Short Answer

Expert verified
The function in vertex form is \(y = -2(x - \frac{3}{4})^2 + \frac{1}{8}\). The vertex is at (\frac{3}{4}, \frac{1}{8}) and the parabola opens downward.

Step by step solution

01

- Identify Coefficients

Identify the coefficients of the quadratic function given. Here, it's given as \(y = -2x^2 + 3x - 1\). The coefficients are: a = -2, b = 3, c = -1.
02

- Complete the Square for the Quadratic Term

Rearrange the quadratic function to help us complete the square. Start from: \(y = -2x^2 + 3x - 1\).Factor out the coefficient of the quadratic term from the x-terms: \(y = -2(x^2 - \frac{3}{2}x) - 1\).
03

- Add and Subtract the Square Term

To complete the square inside the parentheses, we need to add and subtract (\frac{b}{2a})^2. Here, \((-\frac{3}{4})^2 = \frac{9}{16}\). So we rewrite the equation as: \(y = -2(x^2 - \frac{3}{2}x + \frac{9}{16} - \frac{9}{16}) - 1\).
04

- Simplify the Equation

Simplify the equation by moving the subtracted term outside the parentheses: \(y = -2((x - \frac{3}{4})^2 - \frac{9}{16}) - 1\). Distribute the -2 and simplify further: \(y = -2(x - \frac{3}{4})^2 + \frac{18}{16} - 1\), \(y = -2(x - \frac{3}{4})^2 + \frac{9}{8} - \frac{8}{8}\), and final result: \(y = -2(x - \frac{3}{4})^2 + \frac{1}{8}\).
05

- Rewrite in Vertex Form

The function written in the vertex form is: \(y = a(x - h)^2 + k\), where a = -2, h = \frac{3}{4} and k = \frac{1}{8}\.Therefore, the vertex form is: \(y = -2(x - \frac{3}{4})^2 + \frac{1}{8}\).
06

- Sketch the Graph

Now sketch the graph based on the vertex form. The vertex is at (\frac{3}{4}, \frac{1}{8}).Since a = -2, the parabola opens downwards and is narrower than the parent parabola \(y = x^2\). Plotting the points and drawing the graph gives a downward opening parabola.

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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 equation of the form \(y = ax^2 + bx + c\). This equation represents a parabola when graphed. The coefficients determine the shape and position of the parabola:
  • \textbf{a} - Controls the direction and width of the parabola (upward if positive, downward if negative).
  • \textbf{b} - Influences the position and orientation of the parabola.
  • \textbf{c} - Determines the vertical offset of the parabola.
Understanding these coefficients is crucial for transforming the quadratic function into vertex form. The vertex form of a quadratic function is \(y = a(x - h)^2 + k\). This form makes it easier to identify the vertex of the parabola, \((h, k)\).
Completing the Square
Completing the square is a method used to convert a quadratic equation into its vertex form. This technique helps in identifying the vertex and graphing the parabola. Here is a step-by-step process to complete the square: Begin with the quadratic function \(y = ax^2 + bx + c\). For example, take \(y = -2x^2 + 3x - 1\).
  • Identify and factor out the coefficient of the quadratic term: \(y = -2(x^2 - \frac{3}{2}x) - 1\).
  • Add and subtract the square term inside the parentheses to complete the square. This term is \((\frac{b}{2a})^2\), which for our example is \((-\frac{3}{4})^2 = \frac{9}{16}\).
  • Rearrange and then simplify the inside of the parenthesis: \(y = -2(x^2 - \frac{3}{2}x + \frac{9}{16} - \frac{9}{16}) - 1\)
The equation is then simplified to factor out a perfect square trinomial: \(y = -2((x - \frac{3}{4})^2 - \frac{9}{16}) - 1\). Distribute the \(-2\) and simplify further to get \(y = -2(x - \frac{3}{4})^2 + \frac{1}{8}\). Now the quadratic equation is in vertex form.
Parabola Graph
Graphing a quadratic function involves plotting a parabola. When a quadratic function is in vertex form, \(y = a(x - h)^2 + k\), it becomes easy to graph. Begin by identifying the vertex, \((h, k)\), from the vertex form. For our example: \(y = -2(x - \frac{3}{4})^2 + \frac{1}{8}\), the vertex is \((\frac{3}{4}, \frac{1}{8})\). The coefficient \(a\) tells us that the parabola opens downwards since \(a = -2\). It is narrower compared to its parent function \(y = x^2\). Steps to graph:
  • Plot the vertex point \((\frac{3}{4}, \frac{1}{8})\).
  • Determine additional points by choosing x-values around the vertex and calculating corresponding y-values using the vertex form equation.
  • Draw a smooth curve through these points to complete the parabola shape.
The resulting graph gives a clear visual representation of the quadratic function, making it easier to understand its properties.

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