Problem 19 a. Write the function in vertex ... [FREE SOLUTION]

Chapter 3: Problem 19

a. Write the function in vertex form. b. Identify the vertex. c. Identify the $x$ -intercepts. d. Identify the $y$ -intercept. e. Sketch the function. f. Determine the axis of symmetry. g. Determine the minimum or maximum h. State the domain and range. (See Example 2 ) value of the function. $$ p(x)=3 x^{2}-12 x-7 $$

Short Answer

Expert verified

a.$ p(x) = 3(x - 2)^2 - 19 $b. (2, -19)c. $ 2 \pm \sqrt{\frac{19}{3}} $d. (0, -7)e. A parabola opening upwardsaf. x = 2g. Minimum: -19h.Domain:$ (-\infty, +\infty) $, Range:$ [-19, +\infty) $

Step by step solution

Convert to Vertex Form

To convert the quadratic function to vertex form, complete the square. Starting with the given function: \[ p(x) = 3x^2 - 12x - 7 \]First, factor out the coefficient of the quadratic term from the first two terms:\[ p(x) = 3(x^2 - 4x) - 7 \]Next, complete the square inside the parentheses. To do this, take half of the coefficient of the linear term, square it, and add and subtract it inside the parentheses:\[ p(x) = 3(x^2 - 4x + 4 - 4) - 7 \]Simplify the expression inside the parentheses:\[ p(x) = 3((x - 2)^2 - 4) - 7 \]Distribute the 3 and simplify:\[ p(x) = 3(x - 2)^2 - 12 - 7 \]Combine like terms:\[ p(x) = 3(x - 2)^2 - 19 \]Thus, the vertex form of the function is:\[ p(x) = 3(x - 2)^2 - 19 \]

Identify the Vertex

From the vertex form $ p(x) = 3(x - 2)^2 - 19 $, the vertex $(h, k)$ is given by $(2, -19)$. So the vertex is $ (2, -19) $.

Identify the x-Intercepts

To find the $ x $-intercepts, solve for $ x $ when $ p(x) = 0 $.Set the quadratic function to zero:\[ 0 = 3(x - 2)^2 - 19 \]Add 19 to both sides:\[ 19 = 3(x - 2)^2 \]Divide both sides by 3:\[ \frac{19}{3} = (x - 2)^2 \]Take the square root of both sides:\[ x - 2 = \pm \sqrt{\frac{19}{3}} \]Solve for $ x $:\[ x = 2 \pm \sqrt{\frac{19}{3}} \]So the $ x $-intercepts are $ x = 2 + \sqrt{\frac{19}{3}} $ and $ x = 2 - \sqrt{\frac{19}{3}} $.

Identify the y-Intercept

To find the $ y $-intercept, set $ x = 0 $ and solve for $ p(x) $:\[ p(0) = 3(0)^2 - 12(0) - 7 = -7 \]So the $ y $-intercept is $ (0, -7) $.

Sketch the Function

Begin by plotting the vertex $ (2, -19) $, the $ x $-intercepts $ 2 \pm \sqrt{\frac{19}{3}} $, and the $ y $-intercept $ (0, -7) $. Draw the parabola opening upwards since the coefficient of the $ x^2 $-term is positive.

Determine the Axis of Symmetry

The axis of symmetry for a parabola in vertex form $ p(x) = a(x - h)^2 + k $ is given by $ x = h $. Thus, the axis of symmetry is $ x = 2 $.

Determine the Maximum or Minimum Value

Since the parabola opens upwards (the coefficient of $ x^2 $ is positive), it has a minimum value at the vertex. Thus, the minimum value of the function is $ -19 $.

State the Domain and Range

The domain of any quadratic function is all real numbers $ (-\infty, +\infty) $. The range of $ p(x) = 3(x - 2)^2 - 19 $ is $ [-19, +\infty) $ since the minimum value is $ -19 $ and the parabola opens upwards.

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Key Concepts

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

Quadratic Function

A quadratic function is a type of function that follows the form: \[ f(x) = ax^2 + bx + c \] where $ a $, $ b $, and $ c $ are constants, and the graph of this function is a parabola. Parabolas are U-shaped curves that open either upwards or downwards, depending on the value of $ a $.

If $ a > 0 $, the parabola opens upwards.
If $ a < 0 $, it opens downwards.

In the example, we start with the function \[ p(x) = 3x^2 - 12x - 7 \]. By understanding this structure, we can rewrite it in different forms, such as vertex form, to identify key features.

Completing the Square

Completing the square is a technique used to transform a quadratic function into its vertex form: \[ p(x) = a(x-h)^2 + k \]. This form makes it easier to identify the vertex of the parabola. Following these steps helps complete the square:

Factor out the coefficient of the $ x^2 $ term from the first two terms.
Take half of the coefficient of the $ x $ term, square it, and add and subtract it inside the parentheses.
Simplify the expression and distribute any factors.

For our function \[ p(x) = 3x^2 - 12x - 7 \], we completed the square and transformed it into \[ p(x) = 3(x-2)^2 - 19 \]. This new form makes finding the vertex straightforward.

Vertex Identification

The vertex of a quadratic function is the point where the parabola changes direction. It can be a maximum or a minimum depending on whether the parabola opens downwards or upwards. For a function in vertex form $ p(x) = a(x-h)^2 + k $, the vertex is $ (h, k) $. From the given example, the transformed function is \[ p(x) = 3(x-2)^2 - 19 \]. Here, the vertex $ (h, k) $ is easily identified as $ (2, -19) $. This vertex represents the minimum point of the parabola since the parabola opens upwards.

Intercepts

Intercepts are points where the graph intersects the axes. They include:

$ x $-intercepts, where the function crosses the x-axis ($ p(x) = 0 $).
$ y $-intercept, where the function crosses the y-axis ($ x = 0 $).

x-Intercepts

To find the $ x $-intercepts, set the function to zero and solve for $ x $: \[ 0 = 3(x - 2)^2 - 19 \], which gives us \[ x = 2 \pm \sqrt{\frac{19}{3}} \]. So, the $ x $-intercepts are $ x = 2 + \sqrt{\frac{19}{3}} $ and $ x = 2 - \sqrt{\frac{19}{3}} $.

y-Intercept

To find the $ y $-intercept, set $ x = 0 $: \[ p(0) = 3(0)^2 - 12(0) - 7 = -7 \]. Thus, the $ y $-intercept is $ (0, -7) $. These intercepts are crucial for sketching and understanding the behavior of the quadratic function.

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Short Answer

Step by step solution

Convert to Vertex Form

Identify the Vertex

Identify the x-Intercepts

Identify the y-Intercept

Sketch the Function

Determine the Axis of Symmetry

Determine the Maximum or Minimum Value

State the Domain and Range

Key Concepts

Quadratic Function

Completing the Square

Vertex Identification

Intercepts

x-Intercepts

y-Intercept

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