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Chapter 9: Problem 2

Does the parabola open upward or downward? Explain. $$ y=x 2-12 x+32 $$

Short Answer

Expert verified
The parabola opens upward because the coefficient of the x虏 term is positive.

Step by step solution

01

Identify the Quadratic Equation

The given equation of the parabola is in the format of a quadratic equation: y = ax^2 + bx + c. Here, by comparing, we can see that the equation is y = x^2 - 12x + 32, where a = 1, b = -12, and c = 32.
02

Determine the Direction Based on 'a'

The direction in which a parabola opens is determined by the coefficient 'a' of the x^2 term. - If 'a' is positive, the parabola opens upwards. - If 'a' is negative, the parabola opens downwards. In our equation, a = 1, which is positive.
03

Conclusion on the Direction of Opening

Since 'a' is positive in the equation, this indicates that the parabola opens upward.

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Key Concepts

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

Quadratic Equation
A quadratic equation is any equation that can be written in the form \( y = ax^2 + bx + c \), where "a", "b", and "c" are constants, and \( a eq 0 \). The term "quadratic" comes from "quadratus," a Latin word meaning "square". This refers to the \( x^2 \) term, which is essential in defining the equation as quadratic. In any quadratic equation, the highest power of the variable \( x \) is 2. Hence,
  • it features a parabola, a U-shaped curve.
  • the coefficient "a" cannot be zero because otherwise the equation would be linear, not quadratic.
Quadratic equations are used frequently in mathematics to solve problems involving projectile motion, areas of shapes, and many real-world phenomena. They are central to algebra and form the backbone of more advanced mathematical concepts.
Direction of Parabola
The direction in which a parabola opens is an important trait determined by the leading coefficient "a" in the quadratic equation formula \( y = ax^2 + bx + c \). This coefficient influences the orientation and shape of the parabola on a coordinate plane.
- When \( a > 0 \), the parabola opens upwards, resembling a smile or a U-shape.
- Conversely, when \( a < 0 \), the parabola opens downwards, similar to a frown or an upside-down U-shape.
  • The sign of "a" therefore crucially affects the graph's appearance, signifying either a minimum point when the parabola opens upward or a maximum point when it opens downward.
  • In practical contexts, this can help determine the direction of physical actions, such as the highest or lowest point of a thrown object.
Understanding the direction of a parabola is fundamental to mastering quadratic functions and their applications.
Coefficient of \( x^2 \)
The coefficient of \( x^2 \) in the quadratic equation \( y = ax^2 + bx + c \) is called "a". Recognizing this coefficient is essential because it not only determines the parabola's direction but also affects the graph's steepness or "spread".
  • A larger absolute value of "a" means a steeper parabola. The graph narrows, appearing "thinner" as it rises or falls more sharply.
  • Conversely, a smaller absolute value of "a" leads to a wider parabola, with a more gradual slope.
  • The coefficient "a", when greater than zero, ensures the U-shaped graph points upwards, whereas a negative "a" inverts this shape.
In the given example, \( a = 1 \), which results in an upward-opening parabola with moderate width based on the unit value of "a".

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Most popular questions from this chapter

Use the quadratic formula to solve the following. $$x_{2}-1.6 x-0.8=0$$

Use the quadratic formula to solve the following. $$t 2-4 t-1=0$$

Graph. Find the vertex and the y-intercept. In addition, find the x-intercepts if they exist. $$ y=-x 2-6 x+7 $$

Solve by completing the square and round off the solutions to the nearest hundredth. $$(6 x+1) 2-6(6 x+1)=0$$

Solve by factoring and then solve by completing the square. Check answers. $$x 2+2 x-8=0$$
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