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Chapter 4: Problem 27

(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. \(f(x)=2(x-6)^{2}+3\)

Short Answer

Expert verified
Vertex: (6, 3). Axis of symmetry: x = 6. The graph is concave up.

Step by step solution

01

- Identify the Standard Form

The given quadratic function is in the form: y = a(x-h)^2 + kwhere (h, k) is the vertex. Here, a = 2, h = 6, and k = 3.
02

- Find the Vertex

Using the identified values from the standard form, the vertex (h, k) is (6, 3).
03

- Determine the Axis of Symmetry

The axis of symmetry in a quadratic function of the given form is x = h. So, the axis of symmetry is x = 6.
04

- Determine Concavity

Since the coefficient of \(x^2\), which is 'a', is positive (a=2), the graph of the quadratic function is concave up.
05

- Graph the Function

Plot the vertex (6, 3) on a coordinate plane. Draw the axis of symmetry as a vertical line at x=6. Since the graph is concave up, sketch the parabola opening upwards around the vertex.

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Key Concepts

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

Vertex
The vertex of a quadratic function is a crucial point on its graph. For the function given, \(f(x) = 2(x-6)^2 + 3\), we can identify the vertex using the standard form equation: \(y = a(x-h)^2 + k\). Here, The vertex (h, k) of the function is (6, 3). The vertex represents the highest or lowest point on the graph, depending on the graph's direction. In this case, the vertex means the lowest point since the parabola opens upwards.
Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex of the quadratic function and splits the parabola into two mirror images. For the function \(f(x) = 2(x-6)^2 + 3\), we find that the axis of symmetry is given by the equation \(x = h\). Using the value of h = 6 from our equation, the axis of symmetry is This means that if you were to fold the graph along the line x = 6, both sides of the parabola would match perfectly.
Concavity
Concavity describes the direction in which the parabola opens. This is determined by the coefficient 'a' in our standard form equation \(y = a(x-h)^2 + k\). In the given function \(f(x) = 2(x-6)^2 + 3\), the coefficient 'a' is 2. Because The parabola is concave up, meaning it opens upwards. If 'a' were negative, the parabola would be concave down, opening downwards. Understanding the concavity helps us predict the shape and direction of the parabola in the graph.
Graphing Parabolas
Graphing a parabola involves a few careful steps. Let鈥檚 graph \(f(x) = 2(x-6)^2 + 3\). Start with the vertex, which we identified as Next, draw the axis of symmetry: a vertical line at Because the parabola is concave up (as 'a' is positive), sketch the parabola opening upwards around the vertex. It might also be useful to plot additional points on either side of the vertex to ensure more accuracy in the shape. Select some values for x near the vertex, calculate the corresponding y values, and plot these points. Connecting these points smoothly will give you the graph of the quadratic function.

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Most popular questions from this chapter

Use the fact that a quadratic function of the form \(f(x)=a x^{2}+b x+c\) with \(b^{2}-4 a c>0\) may also be written in the form \(f(x)=a\left(x-r_{1}\right)\left(x-r_{2}\right),\) where \(r_{1}\) and \(r_{2}\) are the \(x\) -intercepts of the graph of the quadratic function. (a) Find quadratic functions whose \(x\) -intercepts are -3 and 1 with \(a=1 ; a=2 ; a=-2 ; a=5\) (b) How does the value of \(a\) affect the intercepts? (c) How does the value of \(a\) affect the axis of symmetry? (d) How does the value of \(a\) affect the vertex? (e) Compare the \(x\) -coordinate of the vertex with the midpoint of the \(x\) -intercepts. What might you conclude?

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=4 x^{2}-4 x\)

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

Suppose that the quantity supplied \(S\) and the quantity demanded \(D\) of hot dogs at a baseball game are given by the following functions:$$\begin{array}{l}S(p)=-2000+3000 p \\\D(p)=10,000-1000 p\end{array}$$ where \(p\) is the price of a hot dog. (a) Find the equilibrium price for hot dogs at the baseball game. What is the equilibrium quantity? (b) Determine the prices for which quantity demanded is less than quantity supplied. (c) What do you think will eventually happen to the price of hot dogs if quantity demanded is less than quantity supplied?

(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 \(f(x)<0\) \(f(x)=-4 x^{2}-6 x+2\)
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