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Chapter 8: Problem 65

Write the equation \(y=\frac{1}{2} x^{2}+4 x-9\) in the form \(y=a(x-h)^{2}+k\).

Short Answer

Expert verified
The equation in the form \(y = a(x-h)^{2} + k\) is \[ y = \frac{1}{2} (x + 4)^{2} - 17 \]

Step by step solution

01

- Identify the given quadratic equation

The given quadratic equation is \[ y = \frac{1}{2} x^{2} + 4x - 9 \]
02

- Factor out the coefficient of x虏

Factor out the coefficient \(\frac{1}{2}\) from the first two terms: \[ y = \frac{1}{2} \left( x^{2} + 8x \right) - 9 \]
03

- Complete the square

Complete the square inside the parentheses. Take half of the coefficient of x (which is 8), square it, and add it inside the parentheses while compensating outside: \[ y = \frac{1}{2} \left( x^{2} + 8x + 16 \right) - 9 - \frac{1}{2} (16) \]
04

- Simplify the equation

Simplify the expression to get the form \( y = a(x-h)^{2} + k \): \[ y = \frac{1}{2} (x + 4)^{2} - 9 - 8 \] \[ y = \frac{1}{2} (x + 4)^{2} - 17 \]

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Key Concepts

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

quadratic equations
Quadratic equations are fundamental in algebra and have the general form \( ax^2 + bx + c = 0 \). Here, the highest exponent is 2, which creates a parabola when graphed. Quadratic equations arise in various real-world contexts, such as projectile motion and area calculations. Notice that in the given equation \( y = \frac{1}{2} x^2 + 4x - 9 \), the coefficient \(a = \frac{1}{2} \), \(b = 4\), and \(c = -9\). Understanding the basic structure of quadratic equations is crucial for making sense of their transformations.
vertex form
The vertex form of a quadratic equation helps to easily identify the vertex of the parabola, enhancing our understanding of its graphical representation. The vertex form is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. The conversion of a standard form quadratic equation to vertex form involves completing the square. This method restructures the equation to reveal the vertex's coordinates directly. For our example, the equation \( y = \frac{1}{2} x^2 + 4x - 9 \) was transformed to vertex form:\( y = \frac{1}{2} (x + 4)^2 - 17 \), which indicates that the vertex is at \(( -4, -17 )\).
transformations
Transformations of quadratic equations involve translations, reflections, stretches, and compressions. These transformations shift the position and shape of the parabola in the coordinate plane. For the given quadratic equation undergoing transformation to vertex form, you can observe the following:
  • Horizontal Shift: The term \(x + 4\) indicates a shift 4 units to the left. The positive sign means the shift goes in the negative direction.
  • Vertical Shift: The constant term \(-17\) in the vertex form signals a downward shift by 17 units.
  • Vertical Stretch/Compression: The coefficient \(\frac{1}{2}\) compresses the parabola vertically, making it wider than the standard parabola \(y = x^2\).
Understanding these transformations helps you anticipate how changes in the equation impact the graph.

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Most popular questions from this chapter

Use a graphing calculator to graph the equation corresponding to each inequality. From the display of the graphing calculator, locate one ordered pair in the solution set to the system and check that it satisfies all inequalities of the system. Answers may vary. $$\begin{aligned}&y>e^{x-0.8}\\\&y<\log (x+2.5)\end{aligned}$$

Solve each system of inequalities. $$\begin{array}{r}x \geq 0 \\\y \geq 0 \\\x+y \leq 4\end{array}$$

Find the amount after 3 years and 6 months for an investment of 20,000 dollars that returns \(5 \%\) compounded continuously.

Solve the system \(5 x-9 y=12\) and \(18 y-10 x=20\).

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