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Chapter 4: Problem 23

(a) find the vertex and axis of symmetry of each quadratic function. (b) Determine whether the graph is concave up or concave down. (c) Graph the quadratic function. \(f(x)=(x-3)^{2}-2\)

Short Answer

Expert verified
Vertex: (3, -2), Axis of Symmetry: x = 3, Concave Up.

Step by step solution

01

Identify the Form of the Quadratic Function

Notice that the quadratic function is given in vertex form, which is \[ f(x) = a(x-h)^2 + k \]. Here, we can identify that \( a = 1 \), \( h = 3 \), and \( k = -2 \).
02

Find the Vertex

Using the identified values, the vertex of the function is at the point \( (h, k) = (3, -2) \).
03

Determine the Axis of Symmetry

The axis of symmetry for a quadratic function in vertex form is given by the line \( x = h \). Therefore, the axis of symmetry is \( x = 3 \).
04

Determine Concavity

The sign of \( a \) determines if the graph is concave up or concave down. If \( a > 0 \), the graph is concave up. If \( a < 0 \), the graph is concave down. Here, \( a = 1 \), which is positive, so the graph is concave up.
05

Graph the Quadratic Function

Plot the vertex at \( (3, -2) \) and draw the axis of symmetry at \( x = 3 \). Since the graph is concave up, sketch the parabola opening upwards.

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Key Concepts

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

concavity
Concavity refers to the direction in which the parabola opens. The sign of the leading coefficient \a\ in the vertex form \ f(x) = a(x - h)^2 + k\ determines the concavity: \ a > 0 \ means the parabola opens upwards (concave up), and \ a < 0 \ means it opens downwards (concave down).

In the given problem, \ a = 1 \, which is greater than 0. Therefore, the graph of the quadratic function \f(x) = (x - 3)^2 - 2\ is concave up. Visualize this as the parabola looking like a smile. Using this information, we can confidently sketch the graph by ensuring the curve opens upwards starting from the vertex \(3, -2\).

Understanding concavity is crucial for predicting the general shape and direction of the parabola, aiding in accurate graph plotting.

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Most popular questions from this chapter

A linear function is given. (a) Find the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph each function. (c) What is the average rate of change of each function? (d) Determine whether each function is increasing, decreasing, or constant. $$ h(x)=-\frac{2}{3} x+4 $$

(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 \(f(x)<0\) \(f(x)=4 x^{2}-2 x+1\)

What is the only type of function that has a constant average rate of change? (a) linear function (b) quadratic function (c) step function (d) absolute value function

The data at the top of the next column represent the atmospheric pressure \(p\) (in millibars) and the wind speed \(w\) (in knots) measured during various tropical systems in the Atlantic Ocean. (a) Use a graphing utility to draw a scatter plot of the data, treating atmospheric pressure as the independent variable (b) Use a graphing utility to find the line of best fit that models the relation between atmospheric pressure and wind speed. Express the model using function notation. (c) Interpret the slope. $$ \begin{array}{|cc|} \hline \begin{array}{c} \text { Atmospheric Pressure } \\ \text { (millibars), } \boldsymbol{p} \end{array} & \begin{array}{c} \text { Wind Speed } \\ \text { (knots), } \boldsymbol{w} \end{array} \\ \hline 993 & 50 \\ \hline 994 & 60 \\ \hline 997 & 45 \\ \hline 1003 & 45 \\ \hline 1004 & 40 \\ \hline 1000 & 55 \\ \hline 994 & 55 \\ \hline 942 & 105 \\ \hline 1006 & 40 \\ \hline 942 & 120 \\ \hline 986 & 50 \\ 983 & 70 \\ \hline 940 & 120 \\ \hline 966 & 100 \\ \hline 982 & 55 \\ \hline \end{array} $$ (d) Predict the wind speed of a tropical storm if the atmospheric pressure measures 990 millibars. (e) What is the atmospheric pressure of a hurricane if the wind speed is 85 knots?

A linear function is given. (a) Find the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph each function. (c) What is the average rate of change of each function? (d) Determine whether each function is increasing, decreasing, or constant. $$ g(x)=5 x-4 $$
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