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Chapter 1: Problem 15

For parts \((a)-(d)\) use the following cquation: \(y=-2 x^{2}\) a. If \(x=0\), find the value of \(y\). b. If \(x\) is greater than zero, what can you say about the value of \(\mathrm{y}^{\prime} ?\) c. If \(x\) is a negative number, what can you say about the value of \(y ?\) d. Can you find an ordered pair that represents a solution to the equation when \(y\) is greater than zero? If so, find it; if not, explain why.

Short Answer

Expert verified
a) y = 0; b) y < 0; c) y < 0; d) y cannot be greater than zero

Step by step solution

01

Find the value of y when x = 0

To find the value of y when x = 0, substitute x = 0 into the equation. The equation is given as \(y = -2x^2\). Substitute \(x = 0\) to get: \[y = -2(0)^2 = 0\] Thus, the value of y is 0.
02

Determine the value of y if x > 0

If \(x\) is greater than zero, we need to determine the nature of y. Since the equation is \(y = -2x^2\), for any positive value of \(x\), \(x^2\) will be positive. Multiplying a positive number by -2 will result in a negative value. Therefore, when \(x > 0\), \(y < 0\).
03

Determine the value of y if x < 0

If \(x\) is a negative number, we follow a similar logic. The equation is \(y = -2x^2\). For any negative value of \(x\), \(x^2\) will still be positive because the square of any real number is positive. Multiplying a positive number by -2 results in a negative value. Therefore, when \(x < 0\), \(y < 0\).
04

Analyze if y can be greater than zero

To determine if \(y\) can be greater than zero, we analyze the equation \(y = -2x^2\). The term \(-2x^2\) will always be less than or equal to zero because the product of a positive number (\(x^2\)) and -2 is always non-positive. Thus, \(y\) cannot be greater than zero. It is either negative or zero.

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Key Concepts

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

Solving Quadratic Equations
Quadratic equations are of the form \(y = ax^2 + bx + c\). In our exercise, we have the equation \(y = -2x^2\). To solve quadratic equations, follow these steps:
  • Identify the coefficients \(a, b,\) and \(c\). In our example, \(a = -2\), \(b = 0\), and \(c = 0\).
  • If necessary, use methods like factoring, completing the square, or the quadratic formula \([x = \frac{-b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}]\).
In some cases, simple substitution is enough, as seen when \(x = 0\) gives \(y = 0\).
For positive \(x\), we substitute \(x\) into the equation to find \(y\). Each positive \(x\) results in a negative \(y\) value because of the negative coefficient \(-2\).
Similarly, substituting negative \(x\), yields \(y < 0\). Critical steps include:
  • Analyzing the sign of \(a\).
  • Understanding the implications of squaring any real number \(x^2 \geq 0\).
  • Multiplying by a negative coefficient to find the resulting \(y\).
Properties of Quadratic Functions
Quadratic functions have unique properties that can be easily identified by their standard form \(y = ax^2 + bx + c\). Here are key properties using our example \(y = -2x^2\):
  • The parabola opens downward because \(a = -2 < 0\).
  • The vertex represents the maximum point. Here, the vertex is at \((0, 0)\).
  • The axis of symmetry, a vertical line passing through the vertex, is \(x = 0\).
  • The function's domain is all real numbers, but the range here is \(( - \infty, 0]\).
These properties help in sketching and understanding the function's behavior.
When \(x = 0\), \(y = 0\), leading to the vertex at the origin. Regardless of positive or negative \(x\), \(y < 0\), demonstrating that the function does not achieve positive \(y\) values.
Inequalities in Quadratic Equations
Understanding inequalities within quadratic equations helps decipher the possible values of \(y\). For the given equation \(y = -2x^2\), consider these points:
  • For \(x > 0\), \(y < 0\) because squaring \(x\) (always positive) and multiplying by a negative coefficient \(-2\) results in a negative number.
  • For \(x < 0\), the scenario remains that \(y < 0\) since \(x^2\) is positive and \(-2 \cdot \ positive = negative\).
The equation \(y = -2x^2\) implies that \(y \leq 0\). In simpler terms:
  • \

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