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Chapter 8: Problem 36

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples 1 through 4 . $$ f(x)=x^{2}-6 x+11 $$

Short Answer

Expert verified
The vertex is \((3, 2)\), the graph opens upward, the y-intercept is \((0, 11)\), and there are no real x-intercepts.

Step by step solution

01

Identify the quadratic function form

The given function is \( f(x) = x^2 - 6x + 11 \), which is in the standard form \( ax^2 + bx + c \) where \( a = 1 \), \( b = -6 \), and \( c = 11 \).
02

Determine whether the parabola opens up or down

The parabola opens upwards because the coefficient \( a = 1 \) is positive.
03

Find the vertex using vertex formula

The vertex formula for a quadratic function in standard form is \( x = \frac{-b}{2a} \). Substitute \( a = 1 \) and \( b = -6 \) into the formula: \[ x = \frac{-(-6)}{2 \times 1} = \frac{6}{2} = 3 \]To find the \( y \)-coordinate of the vertex, substitute \( x = 3 \) back into the original function: \[ f(3) = 3^2 - 6 \times 3 + 11 = 9 - 18 + 11 = 2 \]Thus, the vertex is \( (3, 2) \).
04

Find the y-intercept

The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the function: \[ f(0) = 0^2 - 6 \times 0 + 11 = 11 \]Thus, the y-intercept is \( (0, 11) \).
05

Check for x-intercepts

The x-intercepts occur where \( f(x) = 0 \). This requires solving the equation \[ x^2 - 6x + 11 = 0 \]Calculate the discriminant \( b^2 - 4ac = (-6)^2 - 4 \times 1 \times 11 = 36 - 44 = -8 \).Since the discriminant is negative, there are no real x-intercepts.
06

Sketch the graph

The graph is a parabola that opens upwards with vertex at \((3, 2)\) and a y-intercept at \((0, 11)\). Since it has no real x-intercepts, it does not cross the x-axis. Sketch accordingly.

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Key Concepts

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

Parabola Opens Upwards
When you're faced with a quadratic function in the form of \( f(x) = ax^2 + bx + c \), the coefficient \( a \) plays a crucial role in determining the direction of the parabola. If \( a \) is positive, you can be assured the parabola opens upwards, resembling a U shape. For instance, in the function \( f(x) = x^2 - 6x + 11 \), the coefficient \( a = 1 \) is greater than zero.
  • This tells us the parabola goes upwards, which means the vertex will be at the lowest point on the graph.
  • Understanding which way the parabola opens helps in understanding the basic behavior of the function.
Remember, a positive \( a \) value signals a concave up parabola - like a happy face!
Quadratic Function Form
The given quadratic function \( f(x) = x^2 - 6x + 11 \) is in the standard form \( ax^2 + bx + c \) where \( a = 1 \), \( b = -6 \), and \( c = 11 \). This standard form is often the starting point in figuring out key features of the graph.
  • The values of \( a \), \( b \), and \( c \) are really important in understanding the function's behavior.
  • They help determine not only the direction of the opening but also the position of the vertex, intercepts, and more.
Getting comfortable with identifying these coefficients sets the stage for performing essential calculations, like finding the vertex using the vertex formula.
Vertex Formula
Finding the vertex of a quadratic function can be easily done using the vertex formula for standard form \( ax^2 + bx + c \). This formula is \( x = \frac{-b}{2a} \). Simply substitute the values of \( a \) and \( b \) from the quadratic function into it.
  • For \( f(x) = x^2 - 6x + 11 \), \( a = 1 \) and \( b = -6 \); so, \( x = \frac{-(-6)}{2 \times 1} = \frac{6}{2} = 3 \).
  • To find the y-coordinate, plug \( x = 3 \) back into the original equation to get \( f(3) = 2 \).
  • This calculation shows the vertex is \((3, 2)\).
The vertex tells us the graph's highest or lowest point, depending on the direction it opens. It's an essential aspect of sketching the graph accurately.
Discriminant Calculation
The discriminant is a key part of quadratic functions for finding the number and type of roots. For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is given by \( b^2 - 4ac \). Calculating the discriminant will reveal whether the equation has real or complex roots.
  • In the function \( x^2 - 6x + 11 \), the discriminant is \((-6)^2 - 4 \times 1 \times 11 = 36 - 44 = -8\).
  • Since the discriminant is negative, there are no real roots, indicating the parabola doesn't cross the x-axis.
  • This information is crucial in confirming there are no x-intercepts for this graph.
Understanding the discriminant helps to know not only the roots but also to check the intercepts and overall graph features.

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Most popular questions from this chapter

Without calculating, tell whether each graph has a minimum value or a maximum value. See the Concept Check in the section. $$ f(x)=3-\frac{1}{2} x^{2} $$

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples 1 through 4 . $$ f(x)=x^{2}-2 x-15 $$

Without solving, determine whether the solutions of each equation are real numbers or complex but not real numbers. See the Concept Check in this section. $$ (2 y-5)^{2}+7=3 $$

In your own words, explain how to calculate the number that will complete the square on an expression such as \(x^{2}-5 x\)

Simplify each expression. See Section 7.1 $$ \frac{1}{2}-\sqrt{\frac{9}{4}} $$
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