Lesson 23: Using Quadratic Expressions in
Vertex Form to Solve Problems
Cool Down: Looking for The Greatest or the Least
1. Without graphing, find the vertex of the graph of a quadratic function defined by
1²-14x-60. Show your reasoning.
2. Does the y-coordinate of the vertex correspond to a maximum or a minimum value
of the function? Explain how you know.

Answer:

25 cups

Step-by-step explanation:

6 1/8 quarts is equal to 25 cups.

[tex]6\frac{1}{8}[/tex] quarts is equal to 49/2 cups.

Given that we have [tex]6\frac{1}{8}[/tex] quarts, we can multiply this value by 4 to obtain the equivalent amount in cups:

[tex]6\frac{1}{8}[/tex] quarts × 4 cups per quart

= (6 + 1/8)× 4 cups

First, let's convert the mixed number [tex]6\frac{1}{8}[/tex] to an improper fraction:

[tex]6\frac{1}{8}[/tex] = (6 × 8 + 1) / 8

= 49/8

Now, (49/8) × 4 cups

= (49 × 4) / 8 cups

= 196/8 cups

Divide both the numerator and denominator by their greatest common divisor, which is 4:

(196/8) ÷ 4/4 = 49/2

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Answer:

x = 12

Step-by-step explanation:

3/9 = 4/x

3x = 36

x = 12

Answer:

The quadrilateral ABCD was dilated with a scale factor of 1.8

Take the value of x before the dilation (4), and multiply that by 1.8

The result should be 7.2

Hope this helps.

To evaluate the expression 5/6 - 3/4, we need to first convert the fractions to a common denominator. The least common multiple of 6 and 4 is 12, so we can rewrite the fractions as follows:

5/6 - 3/4 = (5/6)(2/2) - (3/4)(3/3)

= 10/12 - 9/12

= 1/12

Therefore, Devon used 1/12 of a cup more sugar than flour.

Answer: -2x + 3y > 7

Step-by-step explanation:

x coordinate is -2

y coordinate is 3

> is the left side

Besides that answer is the only one that makes sense if you only look at the x and y coordinate

The total cost of the dinner is given by the equation A = $ 259.457

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the total cost of the dinner be represented as A

Now , the equation will be

The restaurant bill is = $ 235.87

The percentage of tip for the waitress= 20 %

So , the amount of tip = restaurant bill x percentage of tip for the waitress

Substituting the values in the equation , we get

The amount of tip = 235.87 x ( 20/100 )

On simplifying the equation , we get

The amount of tip = $ 47.174

So , the restaurant bill including the tip = $ 235.87 + $ 47.174

The restaurant bill including the tip = $ 283.044

Now , you have a coupon for 10% off the price of food

So , Cost of dinner with the coupon discount = restaurant bill including the tip  - ( 10/100 )$ 235.87

On simplifying the equation , we get

Cost of dinner with the coupon discount = 283.044 - 23.587

Cost of dinner with the coupon discount = $ 259.457

So , the total cost of dinner A = $ 259.457

Therefore , the value of A is $ 259.457

Hence , the cost of dinner is $ 259.457

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Two-dimensional cross-sections could be obtained from a right rectangular prism a Rectangles. Option D

Generally, A right rectangular prism may give rise to three distinct varieties of two-dimensional cross-sections, and they are as follows:

A rectangle may be created by cutting the right rectangular prism along either of its faces to create a parallelogram, which can then be folded in half.

Obtaining a parallelogram requires cutting the correct rectangular prism at an angle that is not perpendicular to any of its faces. This will result in the formation of a parallelogram.

You may acquire a trapezoid by cutting the right rectangular prism at an angle that is not parallel to any of its faces. This will result in the formation of a trapezoid.

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The amount of money loaned at a rate of 7% will be $7,000.

The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.

A bank loaned out $10,000, part of it at the rate of 7% per year and the rest at 17% per year. If the interest received in one year totaled $1000.

Let 'x' be the amount of money loaned at a rate of 7%. Then the amount of money loaned at a rate of 17% will be ($10,000 - x). Then the equation is given as,

0.07x + 0.17($10,000 - x) = $1,000

0.07x - 0.17x + $1,700 = $1,000

0.10x = $700

x = $7,000

The amount of money loaned at a rate of 7% will be $7,000.

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The matching statements and reasons:

Answer:

1) 7 pizzas

2) 20 people

Step-by-step explanation:

8 people for 2 pizzas is

4 people for 1 pizza

[tex]\frac{1}{4}[/tex] = [tex]\frac{x}{28}[/tex]

4 x 7 = 28

1 x 7 = 7

She will need to order 7 pizzas for 28 people.

1 pizza is needed for 4 people

5 pizzas will allow 20 people (5 x 4)

Answer:

Each person would have to get 8.3 oz of the drink.

Step-by-step explanation:

100/12=8.3

(Sorry if I misunderstood your question)

The ratio of the base areas and ratios of volume are respectively; 1/9 and 1/27

A) Base area of the cone has the formula;

A_b = 3V/h

where;

V is volume

h is the vertical height

Since the ratio of both heights is 5:15 = 1:3, then;

Ratio of base areas = 1²/3² = 1/9

B) Formula for the volume of a cone is;

V = ¹/₃πr²h

Thus;

Va = ¹/₃π(r²)5

Va = ⁵/₃π(r²)

Vb = ¹/₃π(R²)15

Thus;

Va/Vb = ⁵/₃π(r²)/¹/₃π(R²)15

Earlier, we saw that;

r²/R² = 1/9. Thus;

Va/Vb = (1/3) x (1/9)

Va/Vb = 1/27

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Answer:

x = -1

Step-by-step explanation:

There are lots of places to make mistakes.  I think that I did it correctly.

The length of the railing that was parallel to the steps is; 6.65 meters

Let us first of all convert the distance in meters to centimeters. From unit conversion, we know that 1 meter equals 100 centimeters. Thus;

3.6m * 100cm / 1m = 360cm

Now we calculate the number of steps as;

360 cm / 18 cm = 20 steps

Since each step measures 28 cm, it means that total measure of steps is;

20 * 28 = 560 cm

To calculate the length of the ladder, we must apply Pythagoras theorem: c² = a² + b²

From the question, we can say that;

a = 360; b = 560

Thus;

c² = 360² + 560²

c² = 443200

c = √443200

c = 665.73

Expressing 665.73 cm in meters would give us 6.65 meters.

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Answer:

Do you have answer choices?

The possible answers are Plane PLN, Plane LNR, Plane NRP, and Plane RPL.

Rectangle measures 6 by 4 whereas rectangle b measures 6 by 2. A rectangle has two dimensions: a length and a width that are both perpendicular to it.

How to calculate a rectangle's dimensions: The perimeter equation can be expressed in terms of one of the following dimensions as follows: P = 2(a+b) b = P/2−a The equation for the area (A) plus the following one is as follows: The equation for area is A = ab.

We are given a square of 36 square centimeters.

The square's size will be 6 cm as a result.

Two rectangles are created by dividing the square into two equal-sized pieces.

Let the rectangles' measurements be 6*a and 6*b. Consequently, we can establish the ratio.

Moreover, we can set the combined area of rectangles equal to the area of square to form another equation:

Using substitution method, we can solve for 'a' and 'b' as shown below:

And

Therefore, dimensions of the two rectangles are 6x4 and 6x2, respectively.

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The values of the expressions when x = -1 will be;

(i) 1

(ii) 0

(iii) 1

Algebraic expressions are the idea of expressing numbers using letters or alphabets without specifying their actual values. The basics of algebra shows how to express an unknown value using letters such as x, y, z, etc. These letters are called here as variables. An algebraic expression can be a combination of both variables and constants. Any value that is placed before and multiplied by a variable is a coefficient.

when x = -1

(i) -x + 2

-1 + 2 = 1

(ii) x²+ 2x + 1

(-1)² + 2(-1) + 1

1 - 2 + 1 = 0

(iii) 2x² - x - 2

2(-1)² - (-1) - 2

2(1) + 1 - 2

2 + 1 - 2

3 - 2 = 1

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An answer which classifies the function using finite differences is: B. the function is linear with a constant first difference of 3.

Based on the data points in the table, we have the following input and output values;

x                 f(x)

-16            -13.5

-12            -10.5

-8              -7.5

-4              -4.5

0                -1.5

Next, we would determine the slope of this function for data points 1 and 3 as follows;

Slope, m = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope, m = (y₂ - y₁)/(x₂ - x₁)

Slope, m = (-7.5 - (-13.5))/(-8 - (-16))

Slope, m = 3/4

For data points 1 and 5, the slope is given by;

Slope, m = (-1.5 - (-13.5))/(0 - (-16))

Slope, m = 3/4

For the common difference, we have:

Common difference = -10.5 - (-13.5) = 3

Common difference = -7.5 - (-13.5) = 3

Common difference = -4.5 - (-7.5) = 3

Common difference = -1.5 - (-4.5) = 3

Therefore, this function is linear because the slope is constant and the common difference is constant between successive data points.

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Complete Question:

Consider the table that includes the input and output values of a function.

-16    -13.5

-12    -10.5

-8      -7.5

-4      -4.5

0       -1.5

Which answer classifies the function using finite differences?

The function is linear with a constant first difference of –3.

The function is linear with a constant first difference of 3.

The function is quadratic with a constant second difference of –3.

The function is quadratic with a constant second difference of 3.

The domain of the function y=√x+6-7 is x>= -6.

The collection of all potential inputs for a function is its domain. For instance, the domain of f(x)=x2 and g(x)=1/x are all real integers with the exception of x=0.

The set of inputs that a function will accept is known as the domain of the function in mathematics. It can also be written as dom (f) or dom f, where f stands for the function.

To put it more exactly, the domain of f is X given a function f: X - Y. It should be noted that in contemporary mathematical terminology, a function's domain is a component of its definition rather than a quality.

Given data :

The given function is a radical function.

Note that a radical function is only defined for values that will make the radicand greater than or equal to zero.

Hence, the radicand of the given function must be greater than or equal to zero or:

x+6>=0

x + 6 - 6  0 - 6

x>= -6

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Answer:

the slope is 9

Step-by-step explanation:

The slope is the number before the x.

Answer:

Slope = 9

Step-by-step explanation:

Now we have to,

→ find the slope of the equation.

Formula we use,

→ y = mx + b

→ m = Slope

Then the required slope will be,

→ y = mx + b

→ y = 9x + 0

→ y = 9x

→ [ m = 9 ]

Hence, the value of slope is 9.

Alyssa worked 1 hour walking dogs and 13 hours clearing tables. This solution satisfies all of the given conditions.

The first inequality represents the condition that Alyssa must work at most 3 hours walking dogs:

a ≤ 3

The second inequality represents the condition that Alyssa can work a maximum of 15 total hours:

a + y ≤ 15

The third inequality represents the condition that Alyssa must earn at least $130 in a given week:

8a + 10y ≥ 130

We can graph these inequalities on the same coordinate plane to find the region of possible solutions. The solution to the system of inequalities is the region of the plane that satisfies all of the inequalities.

The red line represents the first inequality, a ≤ 3. The blue line represents the second inequality, a + y ≤ 15. The green line represents the third inequality, 8a + 10y ≥ 130.

The region of the plane that satisfies all three inequalities is the area above the red line and below the green line, and to the left of the blue line. This region is shaded in the graph above.

Thus, one possible solution is point (1,13), which is the intersection of the red and blue lines. This corresponds to Alyssa working 1-hour walking dogs and 13 hours clearing tables. This solution satisfies all of the given conditions.

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The density of the wood weighing 19.5061 grams is approximately 19.313 g/cm².

Density is expressed mathematically as;

p = m / v

Where m is mass and v is volume.

Given the data in the question

First, we determine the volume of the wood which is the change in the volume of volume in the cylinder.

Volume of the wood = Final volume of water - Initial volume water

Volume of the wood = 9.76cm³ - 8.75cm³

Volume of the wood = 1.01 cm³

Now, we determine the density of the wood.

p = m / v

p = 19.5061g / 1.01 cm³

p = 19.313 g/cm²

Therefore, the density of the wood is 19.313 g/cm².

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The fruit salad that will taste more melody when the percent of melon in both Arturo and Nachelle are compared would be Nachelle's fruit salad.

The percentage of a quantity is determined when we divide the value by the total value and then multiply the resultant by 100.

For Nachelle;

Quantity of melon = 6 cups

quantity of apple = 1

Total quantity = 7

The percentage of melon = 6/7× 100

= 600/7 = 85.7%

For Arturo;

Quantity of melon = 10 cups

quantity of apple = 3

Total quantity = 13

The percentage of melon = 10/13 ×100

= 1000/13= 76.9%

Therefore,the fruit salad that will be melony would be Nachelle's fruit salad with the percentage of 85.7 of melon making up the salad.

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The expression that is equivalent to (ab3rd power)4th power in the question is a4th power b 12th power.

Exponential expression is the power to which a given term of mathematical expression is raised to. The value of the power to which a variable is raised is not limited. Sometimes, some laws are required to solve some exponential expressions. An example is the law of indices which can be applied to solve some expressions in a given question with large numbered powers. The law of indices is one of the major law to refer to when discussing power of numbers or variables.

Thus the expression that is equivalent to (ab3rd power)4th power is:

(ab3rd power)4th power = (ab^3)^4

expand the given bracket to have;

         (ab^3)^4 = a^4 * b^(3*4)

                        = a^4b^12

Therefore the expression is equivalent to a^4b^12 i.e a4th power b 12th power.

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Answer:

True.

Step-by-step explanation:

When two lines are crossed by a transversal, the interior angles between them will always add up to 180 degrees, which means they can never be parallel.

In the similar quadrilaterals shown, the value of x is 18.3.

A two-dimensional shape with four sides, four vertices, and four angles is referred to as a quadrilateral. Convex and concave are the two main forms. Convex quadrilaterals can also be divided into a number of subgroups, including trapezoids, parallelograms, rectangles, rhombus, and squares.

Given two quadrilaterals whose all four angles are the same. In this condition ratio of their side are equal

Thus,

22 / 18 = x / 15

x = 18.33

Therefore, the value of the x in the given similar quadrilaterals is 18.3.

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The required interest rate in order for Emma to end up with $7,200 is given as follows:

6.5%.

The balance of an account, using continuous compounding, after t years, is presented by the equation given as follows:

[tex]A(t) = A(0)e^{rt}[/tex]

In which the parameters are given as follows:

The values of the parameters in the context of this problem are given as follows:

A(t) = 7200, A(0) = 3300, t = 12.

Hence the interest rate can be obtained solving the exponential equation as follows:

7200 = 3300e^(12r)

e^(12r) = 7200/3300

e^(12r) = 2.181818

ln(e^(12r)) = ln(2.181818).

(as exponential and ln are inverse operations)

12r = ln(2.181818).

r = ln(2.181818)/12

r = 0.065.

As a percentage, the interest rate is given as follows:

6.5%.

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Answer:

-------------------------------------

The x-intercepts are the roots of the function and according to the graph the roots are:

Hence the function is:

Answer:

D)  f(x) = (x - 1)(x - 2)(x - 3)

Step-by-step explanation:

From inspection of the given graph, the points at which the curve intersects the x-axis are:

Factor Theorem

If f(x) is a polynomial, and f(a) = 0, then (x – a) is a factor of f(x).

According to the factor theorem, as f(x) = 0 when x = 1, x = 2 and x = 3 then (x - 1) and (x - 2) and (x - 3) are factors of the polynomial.

Therefore, the equation of the function is:

The additional information needed to that triangle ADN is congruent to triangle DCO using SAS is

Triangles are said to be congruent when the sides and angles equal in accordance to to the triangle congruency rules

Some of the rules include

Side -  Side - Side = SSS

Side - Angle - Side = SAS

Angle - Side - Angle = ASA

Angle - Angle - Side = AAS

Hypotenuse and one leg = HL

The problem requires the prove by Side - Angle - Side = SAS to ensure that the the triangles are congruent

The SAS says that the corresponding sides and the included angle must be congruent

the angles and side used here include

AN - < BAN - BA

DO - < CDN - CD

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Answer:

x=16

Step-by-step explanation:

right!!

Answer:  -0.04x + 0.09

Step-by-step explanation:

(0.95x - 0.04) - (0.99x - 0.13) = 0.95x - 0.04 - 0.99x + 0.13

                                             = 0.95x - 0.99x + 0.13 - 0.04

                                             = (0.95 - 0.99)x + (0.13 - 0.04)

                                             = -0.04x + 0.09