Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 11: Problem 60
(a) write the equation in standard form and (b) use properties of the standard form to graph the equation. $$ y=3 x^{2}-12 x+7 $$
Short Answer
Expert verified
Equation: \(3x^2 - 12x + 7\). Vertex: (2, -5). Opens upward.
Step by step solution
01
Write the Standard Form
The given equation is already a quadratic equation. To write it in the standard form, compare it with the standard form of a quadratic equation - which is - \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In the given equation - y = 3x^2 - 12x + 7, we have: - - a = 3, b - = -12, and c = 7.
02
Identify the Vertex
For a quadratic function in the form \(ax^2 + bx + c\), the vertex (h, k) can be found using the formulas: - \( h = -\frac{b}{2a}\). Substituting the values of \(a\) and \(b\): - - h = -\frac{-12}{2 \times 3} = 2,- and - - k = y(2) = 3(2)^2 - 12(2) + 7 = -5.
03
Find the Axis of Symmetry
The axis of symmetry for a quadratic equation is the vertical line that passes through the vertex. Therefore, the line is \(x = h\). Substituting the value of \(h\), we get the axis of symmetry: - x = 2.
04
Determine the Direction of the Parabola
Since the coefficient of the \(x^2\) term (which is \(a\)) is positive, the graph of the quadratic equation will be a parabola that opens upward.
05
Find Additional Points to Graph the Parabola
To graph the parabola accurately, find additional points by plugging in \(x\) values into the original equation \(y = 3x^2 - 12x + 7\). For example, choose \(x = 0\) and \(x = 4\):- For \(x = 0\), \(y = 3(0)^2 - 12(0) + 7 = 7\).- For \(x = 4\), \(y = 3(4)^2 - 12(4) + 7 = 7\).
06
Plot the Vertex and Additional Points
Plot the vertex (2, -5), the axis of symmetry \(x = 2\), and the additional points (0, 7) and (4, 7) on the graph. Draw the parabola passing through these points.
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.
Standard Form of a Quadratic Equation
The standard form of a quadratic equation is a foundational concept in algebra. It provides a way to express quadratic equations, making them easier to work with, especially when graphing. The standard form is$$y = ax^2 + bx + c$$ where \(a\), \(b\), and \(c\) are constants. In this form:
- \(a\) is the coefficient of \(x^2\)
- \(b\) is the coefficient of \(x\)
- \(c\) is the constant term
Vertex of a Parabola
In a quadratic equation, the vertex represents the highest or lowest point of the parabola. To find the vertex \((h, k)\), we use the formula: h = -b/(2a). Substituting in our values from the standard form,
- h = -(-12) / (2 * 3) = 2
Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex, making it quite straightforward to find. Using our \(h\) value from the vertex formula, the axis of symmetry is given by x = h. For our equation \(y = 3x^2 - 12x + 7\), since \(h = 2\), the axis of symmetry is
- x = 2
Direction of the Parabola
The direction of the parabola is determined by the sign of the coefficient \(a\) in the standard form. If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards. This affects the graph greatly as it tells us whether the vertex represents a minimum or maximum value. In our equation \(y = 3x^2 - 12x + 7\), the coefficient \(a = 3\), which is positive. Thus:
- The parabola opens upwards
