Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 3: Problem 16
For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=4 x^{2}+3 x-1$$
Short Answer
Expert verified
The vertex form is \(P(x) = 4(x + \frac{3}{8})^2 - \frac{25}{16}\),
and the vertex is \((-\frac{3}{8}, -\frac{25}{16})\).
Step by step solution
01
Complete the Square
To rewrite the quadratic function in the form \(P(x) = a(x-h)^2 + k\), we begin by completing the square. We have \(P(x) = 4x^2 + 3x - 1\).First, factor out the leading coefficient from the terms with \(x\):\[P(x) = 4(x^2 + \frac{3}{4}x) - 1\]Next, complete the square for the expression inside the parentheses. Take half of the coefficient of \(x\) (which is \(\frac{3}{4}\)), square it, and then add and subtract it inside the parentheses:\[P(x) = 4\left(x^2 + \frac{3}{4}x + \left(\frac{3}{8}\right)^2 - \left(\frac{3}{8}\right)^2\right) - 1\]Simplify and reorganize the equation:\[P(x) = 4\left((x + \frac{3}{8})^2 - \left(\frac{3}{8}\right)^2\right) - 1\]
02
Simplify the Expression
Now, simplify the expression further. Calculate \(\left(\frac{3}{8}\right)^2\) and distribute the \(4\):\[P(x) = 4(x + \frac{3}{8})^2 - 4\left(\frac{9}{64}\right) - 1\]This results in:\[P(x) = 4(x + \frac{3}{8})^2 - \frac{9}{16} - 1\]Convert \(-1\) to a fraction with a common denominator:\(-1 = -\frac{16}{16}\).Combine the constants:\[P(x) = 4(x + \frac{3}{8})^2 - \left(\frac{9}{16} + \frac{16}{16}\right)\]This evaluates to:\[P(x) = 4(x + \frac{3}{8})^2 - \frac{25}{16}\]
03
Determine the Vertex
From the equation \(P(x) = 4(x + \frac{3}{8})^2 - \frac{25}{16}\), we see that it matches the form \(P(x) = a(x-h)^2 + k\).The vertex \((h, k)\) can be extracted directly from the equation.Thus, the vertex is \((-\frac{3}{8}, -\frac{25}{16})\).
04
Graph the Function
To graph the quadratic function, start by plotting the vertex point \((-\frac{3}{8}, -\frac{25}{16})\).Since \(a = 4\), the parabola opens upwards and is narrower than the standard parabola \(y = x^2\).Sketch the symmetric curve around the vertex, with the parabola being narrower and opening upwards due to the larger value of \(a = 4\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a method used to transform a quadratic equation into a specific format called the vertex form. This format is very helpful for graphing and analyzing the properties of a quadratic function.
For the function \(P(x) = 4x^2 + 3x - 1\), our first step is to extract the coefficient of the squared term, which is 4 in this case. We factor this out from the first two terms:
For the function \(P(x) = 4x^2 + 3x - 1\), our first step is to extract the coefficient of the squared term, which is 4 in this case. We factor this out from the first two terms:
- \(P(x) = 4(x^2 + \frac{3}{4}x) - 1\)
- Take half of \(\frac{3}{4}\) which gives \(\frac{3}{8}\). Square this value (\((\frac{3}{8})^2\)) and then add and subtract the result inside the parentheses.
- This results in \(P(x) = 4(x^2 + \frac{3}{4}x + \frac{9}{64} - \frac{9}{64}) - 1\)
Vertex Form
The vertex form of a quadratic function is \(P(x) = a(x-h)^2 + k\). It provides a clear view of the parabola's vertex, making it especially useful for graphing. Once we have completed the square in our original problem:
Knowing \(a\) helps determine the direction and width of the parabola.
- We've rewritten the function to \(P(x) = 4(x + \frac{3}{8})^2 - \frac{25}{16}\)
- Here, \(a = 4\), \(h = -\frac{3}{8}\), and \(k = -\frac{25}{16}\).
Knowing \(a\) helps determine the direction and width of the parabola.
Graphing Parabolas
To graph a quadratic function effectively, it's beneficial to start with the vertex form of the equation. For our function \(P(x) = 4(x + \frac{3}{8})^2 - \frac{25}{16}\), the vertex is at \((-\frac{3}{8}, -\frac{25}{16})\).
Graphing steps include:
Graphing steps include:
- Plot the Vertex: Begin by plotting this point which represents the highest or lowest point of the parabola.
- Determine the Parabola's Direction: Since \(a = 4\), we know the parabola opens upwards as \(a > 0\).
- Assess the Width: The larger \(a\) value, in this case, indicates a narrower parabola than \(y = x^2\).
- Draw the Symmetrical Shape: Sketch the curve around the vertex, ensuring it's symmetric along the vertical line through the vertex.
Quadratic Vertex
The vertex of a quadratic function is a significant point as it represents the peak or trough of the parabola. It is defined by the coordinates \((h, k)\) derived directly from the vertex form \(P(x) = a(x-h)^2 + k\). In our problem, the vertex is \((-\frac{3}{8}, -\frac{25}{16})\).
The importance of the vertex includes:
The importance of the vertex includes:
- Highest or Lowest Point: In a parabola that opens upwards, like our example, the vertex is the lowest point.
- Axis of Symmetry: The \(x\)-coordinate of the vertex, \(h\), is also the axis of symmetry. This means the parabola is mirrored on either side of this vertical line.
- Easy Graphing: Knowing the vertex simplifies graphing as it provides a starting reference point.
