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Chapter 9: Problem 1

If the lead coefficient of \(y=a x^{2}+b x+c\) is a positive number, does the graph of the equation open up or open down?

Short Answer

Expert verified
The graph opens upwards.

Step by step solution

01

Identify the Lead Coefficient

In the quadratic equation \(y = ax^2 + bx + c\), the lead coefficient is \(a\).
02

Determine the Sign of the Lead Coefficient

Check if the lead coefficient \(a\) is positive or negative. According to the exercise, \(a\) is a positive number.
03

Interpret the Graph's Opening Direction

For a quadratic equation \(y = ax^2 + bx + c\), if \(a > 0\), the graph of the equation opens upwards. Conversely, if \(a < 0\), the graph opens downwards.
04

Conclusion

Since \(a\) is positive, the graph of the quadratic equation opens upwards.

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Key Concepts

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

Lead Coefficient
In a quadratic equation, such as \(y = a x^{2} + b x + c\), the term with the variable squared, \(a x^{2}\), is the most significant term in determining the graph's shape. The coefficient of this term is called the lead coefficient. The lead coefficient, denoted by \(a\), plays a crucial role in understanding the behavior of the quadratic graph. Recognizing and identifying this coefficient is the first step in predicting the direction in which the graph will open.
Positive Coefficient
When discussing quadratic equations, the sign of the lead coefficient \(a\) is very important. A positive lead coefficient means that \(a\) is greater than zero. Mathematically, this can be written as \(a > 0\). The positivity or negativity of this coefficient dictates whether a parabola (the graph of a quadratic function) opens upwards or downwards. For a positive coefficient, the parabola will always open upwards.
Graph Opens Upwards
When the lead coefficient \(a\) in a quadratic equation \(y = a x^{2} + b x + c\) is positive, the graph of the equation opens upwards. This means that the parabola formed by the quadratic function will have its vertex at the bottom, making a 'U' shape.

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Most popular questions from this chapter

Problem: The vertex of a parabola that opens up is \((4,-5)\). Write the equation of the axis of symmetry. Incorrect

Problem: Use the quadratic formula to solve \(x^{2}-3 x+9=0\) Incorrect Answer: \(x=\frac{-(-3) \pm \sqrt{(-3)^{2}-4(1)(9)}}{2(1)}\) $$ \begin{aligned} &x=\frac{3 \pm \sqrt{9-36}}{2} \\ &x=\frac{3 \pm \sqrt{-27}}{2} \\ &x=\frac{3 \pm \sqrt{9} \sqrt{3}}{2} \\ &x=\frac{3 \pm 3 \sqrt{3}}{2} \end{aligned} $$

\(k^{2}+8=-41\)

\((2-3 i)-(6-9 i)\)

The diameter at basal height of a tree is the diameter about \(3 \mathrm{ft}\) above the ground. The approximate shape of the trunk of a tree above basal height is a cone. The formula for the volume of a cone is \(V=\frac{\pi r^{2} h}{3}\), where \(r\) is the radius and \(h\) is the height. Measured from its basal height, a tree is \(50 \mathrm{ft}\) tall. The diameter of its trunk is \(2.5 \mathrm{ft}\). Find the approximate volume of lumber in cubic feet in this trunk. ( \(\pi \approx 3.14\).) Round to the nearest whole number. (Source: G. John Smith; www.math.bcit.ca, 1997)
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