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Chapter 4: Problem 107

Why is the graph of a quadratic function concave up if \(a>0\) and concave down if \(a<0 ?\)

Short Answer

Expert verified
The concavity depends on the sign of 'a': concave up for \(a>0\), concave down for \(a<0\).

Step by step solution

01

Understand the quadratic function

A quadratic function is typically written as \(f(x) = ax^2 + bx + c\). The term 'a' determines the direction of the parabola's concavity.
02

Analyze the effect of 'a' on concavity

The sign of the coefficient 'a' impacts the concavity of the graph:- If \(a > 0\), the parabola opens upwards, making it concave up.- If \(a < 0\), the parabola opens downwards, making it concave down.
03

Connect calculus to the graph's concavity

The second derivative of \(f(x)\) is \(f''(x) = 2a\). The concavity of a function is determined by the sign of its second derivative:- If \(f''(x) > 0\) (i.e., \(2a > 0\)), the function is concave up.- If \(f''(x) < 0\) (i.e., \(2a < 0\)), the function is concave down.
04

Summarize the result

Therefore, the sign of 'a' directly determines the concavity of the parabolic graph.- For \(a>0\), the graph is concave up.- For \(a<0\), the graph is concave down.

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Key Concepts

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

quadratic functions
A quadratic function is one of the simplest types of polynomial functions and is typically written as \(f(x) = ax^2 + bx + c\). Here, \(a, b\), and \(c\) are constants. The coefficient 'a' is very special because it tells us a lot about the graph of the function. Most importantly, it determines how the graph curves, also known as the parabola's concavity.

When you plot a quadratic function, you get a U-shaped curve called a parabola. Because every quadratic function has a highest or lowest point, this shape is very predictable. If you compare it to a line, which is straight, a quadratic function curves either upwards or downwards.

Understanding the role of 'a' helps you grasp how the graph will look even before plotting it. If you master this, you'll be able to visualize quadratic graphs with ease.
parabola concavity
The concavity of a parabola tells us how it curves: upwards or downwards. This concavity depends entirely on the coefficient 'a' in the quadratic function \(f(x) = ax^2 + bx + c\).

When \(a > 0\), the parabola opens upwards. This means it is concave up and looks like a U. Imagine holding a bowl; it holds water because it's facing up.

When \(a < 0\), the parabola opens downwards. This means it is concave down and looks like an upside-down U. Imagine flipping that bowl upside down; it can't hold water now.

The value of 'a' doesn't just change the direction of the curve; it also affects how 'steep' or 'wide' the parabola is. A larger absolute value of 'a' makes the parabola steeper, whereas a smaller absolute value of 'a' makes it wider.

Understanding concavity helps you determine what kind of solutions and behavior you can expect from the quadratic function.
second derivative
Calculus gives us tools to analyze the behavior of functions more deeply. One of these tools is the second derivative.
The second derivative of a function tells us about its concavity.

For the quadratic function \(f(x) = ax^2 + bx + c\), the first derivative is \(f'(x) = 2ax + b\). The second derivative is \(f''(x) = 2a\).

Here's why this matters:

  • If the second derivative \(f''(x) > 0 (or 2a > 0)\), the graph is concave up. This is because a positive second derivative means the slope is increasing, forming a U-shape.

  • If the second derivative \(f''(x) < 0 (or 2a < 0)\), the graph is concave down. This is because a negative second derivative means the slope is decreasing, forming an upside-down U.


So, by looking at the second derivative, you can easily determine the concavity of the quadratic function. This makes the nature of the parabola clear at a glance.

Learning about the second derivative not only helps with quadratic functions but also broadens your understanding of how functions behave overall.

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Most popular questions from this chapter

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=-5 x^{2}+20 x+3\)

A projectile is fired from a cliff 200 feet above the water at an inclination of \(45^{\circ}\) to the horizontal, with a muzzle velocity of 50 feet per second. The height \(h\) of the projectile above the water is modeled by $$ h(x)=\frac{-32 x^{2}}{50^{2}}+x+200 $$ where \(x\) is the horizontal distance of the projectile from the face of the cliff. (a) At what horizontal distance from the face of the cliff is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the face of the cliff will the projectile strike the water? (d) Graph the function \(h, 0 \leq x \leq 200\). (e) Use a graphing utility to verify the solutions found in parts (b) and (c). (f) When the height of the projectile is 100 feet above the water, how far is it from the cliff?

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=-2 x^{2}+2 x-3\)

True or False If the discriminant \(b^{2}-4 a c=0,\) the graph of \(f(x)=a x^{2}+b x+c, a \neq 0,\) touches the \(x\) -axis at its vertex.

The following data represent the width (in yards) and length (in miles) of various tornadoes. $$ \begin{array}{|cc|} \hline \text { Width (yards), } \boldsymbol{w} & \text { Length (miles), } \boldsymbol{L} \\ \hline 200 & 2.5 \\ \hline 350 & 4.8 \\ \hline 180 & 2.0 \\ \hline 300 & 2.5 \\ \hline 500 & 5.8 \\ \hline 400 & 4.5 \\ \hline 500 & 8.0 \\ \hline 800 & 8.0 \\ \hline 100 & 3.4 \\ \hline 50 & 0.5 \\ \hline 700 & 9.0 \\ \hline 600 & 5.7 \\ \hline \end{array} $$ (a) Draw a scatter plot of the data, treating width as the independent variable. (b) What type of relation appears to exist between the width and the length of tornadoes? (c) Select two points and find a linear model that contains the points. (d) Graph the line on the scatter plot drawn in part (a). (e) Use the linear model to predict the length of a tornado that has a width of 450 yards. (f) Interpret the slope of the line found in part (c).
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