Evaluating Linear Functions

An example of a quadratic function that has a maximum value is the parabola defined by the equation y = -(x-3)^2 + 4.

A quadratic function is a type of polynomial function that can be written in the form y = ax^2 + bx + c, where a, b, and c are constants and x is the variable. The graph of a quadratic function is a parabola, which can either open upward or downward depending on the value of the leading coefficient (a).

An equation is a mathematical statement that asserts the equality of two expressions. It is usually written with an "=" sign between the two expressions, such as y = 2x + 3 or x^2 + y^2 = 1. Equations can be used to model real-world situations, and solving an equation means finding the values of the variables that make the equation true.

We can determine that this function has a maximum value by looking at the leading coefficient of the quadratic term (x^2). In this case, the coefficient is -1, which means that the parabola opens downward. Therefore, since the parabola opens downward, the vertex of the parabola is the highest point of the parabola, which means that it is a maximum point.

Also we can find the vertex of the parabola by using the formula x = -b/2a, in this case x= -0 / 2*-1 = 3/2 =1.5, and using this value in the function, we get the y value which is 4, so the vertex is (1.5,4) which confirms that the function has a maximum value at (1.5,4)

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Answer:

720,000,000,000,000 or 720 trillion picoseconds.

Step-by-step explanation:

1 picosecond = 1 trillionth of a second. Therefore 1 second has 1 trillion picoseconds:

0.000 000 000 001 seconds = 1 picosecond

0.000 000 000 001 * 1,000,000,000,000 seconds = 1 * 1,000,000,000,000 picosecond

1 second = 1,000,000,000,000 picoseconds

Now, one minute is the same as 60 seconds so we multiply the equation by 60:

1 minute = 60 seconds = 60,000,000,000,000 picoseconds

Finally, we need 12 minutes to be converted to picoseconds.

12 minutes = 720 seconds = 720,000,000,000,000 picoseconds.

So therefore, 12 minutes is equal to 720,000,000,000,000 picoseconds or 720 trillion picoseconds.

Hope this helps!!!

Option c: y = -x + 5 represents scatter plot 1, Option g: The relationship is not linear represents scatter plot 2, Option a: y = 3x represents scatter plot 3, and Option f: y = x + 3 represents scatter plot 4.

What is a scatter plot?

The graphs that show the association between two variables in a data set are called scatter plots. It displays data points either on a Cartesian system or a two-dimensional plane.

The given equations are -

a. y = 3x

b. y = 4x - 2

c. y = -x + 5

d. y = -5x

e. y = -2x - 4

f. y = x + 3

g. The relationship is not linear.

The first scatter plot is going from positive y-axis to positive x-axis in a linear form. The graph for the equation that goes from positive y-axis to positive x-axis is y = -x + 5.

The second scatter plot is a curved parabola going and not a linear plot. So, the scatter plot is said to be a non-linear scatter plot.

The third scatter plot is going from positive x-axis to infinity in a linear form. The graph for the equation that goes from x-axis to infinity is y = 3x.

The fourth scatter plot is going from positive y-axis to infinity in a linear form. The graph for the equation that goes from positive y-axis to infinity is y = x + 3.

Therefore, scatter plot 1 is represented by equation y = -x + 5, scatter plot 2 is not linear, scatter plot 3 is represented by equation y = 3x and scatter plot 4 is represented by equation y = x + 3.

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The value of a is [tex]-\frac{12}{5}[/tex] if 2 is a zero of the polynomial [tex]$4 x^2-2 x+5 a$[/tex].

As per the given data, the polynomial is p(x) = [tex]$4 x^2-2 x+5 a$[/tex]

The polynomial is zero polynomial if x is 2

The places where a polynomial becomes zero overall are known as the zeros of the polynomial. The term "zero polynomial" refers to a polynomial with a value of zero (0).

Replace x with 2 we get

Therefore [tex]4(2)^2-2(2)+5 \mathrm{a}=0[/tex]

[tex]\Rightarrow[/tex] 4 (4) - 2 (2)+5 (a) = 0

[tex]& \Rightarrow[/tex] 16 - 4 + 5 (a) = 0

[tex]& \Rightarrow[/tex] 12 + 5 (a) = 0

Add -12 on both sides we get

12 + 5 (a) - 12 = 0 - 12

5 (a) = -12

Divided by 5 into both sides we get

a = [tex]-\frac{12}{5}[/tex]

Therefore the value of a is [tex]-\frac{12}{5}[/tex]

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Answer:

The dairy farm produced 30 + n kg of yoghurt, where n is the number of kilograms of cheese produced.

The dairy farm also produced 4n kg of ice cream, where n is the number of kilograms of cheese produced.

Since the dairy farm produced more kilograms of ice cream than yoghurt, we can write the following inequality:

4n > 30 + n

Subtracting n from both sides gives us:

3n > 30

Dividing both sides by 3 gives us:

n > 10

Therefore, the possible values of n are any value greater than 10, such as 11, 12, 13, 14, and so on.

Step-by-step explanation:

Answer:

x ≈ 47.0°

Step-by-step explanation:

using the cosine ratio in the right triangle.

cos x = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{15}{22}[/tex] , then

x = [tex]cos^{-1}[/tex] ( [tex]\frac{15}{22}[/tex] ) ≈ 47.0° ( to the nearest tenth )

As per the graph, there is no restrictions for the domain but the range has a minimum value at y = - 4.

The domain is any real number:

The range is any real number not less than - 4:

According to the following graph, the for the expression y = 5 is horizontal.

The term graph in math is known as the set of vertices and a binary relation between vertices, adjacency.

Here we have given the expression y = 5.

Now, we have to plot these expression on the graph, then we get the following graph.

Through the given graph, we have identified that the expression makes the horizontal line on the graph.

Here we also know that if the equation of a line is y = 5, then it is a horizontal line.

Here we know that the line is parallel to the x-axis and passes through the point y = 5 at the y-axis.

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The total cost of the market research survey is $1000 (base fee) + $75 (cost per question) + $1000 (cost per person surveyed) = $2075.

The total cost to conduct the market research survey is calculated by adding the base fee, the cost per question, and the cost per person surveyed.

The survey has 15 questions and will be sent to 500 people. The survey has a base fee of $1000, which is a fixed cost regardless of the number of questions or people surveyed.

Additionally, the survey has a cost of $5 per question, which is calculated by multiplying the number of questions by the cost per question.

Lastly, the survey has a cost of $2 per person surveyed, which is calculated by multiplying the number of people surveyed by the cost per person surveyed. By adding all these costs together we get the total cost of the survey $2075 which is a combination of fixed and variable costs.

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On solving the provided question, we can say that - here in this inequality 17[tex]\leq[/tex]2x - 3[tex]\leq[/tex]25

A connection between two expressions or values that is not equal in mathematics is referred to as an inequality. Inequality results from an imbalance, thus. In mathematics, an inequality establishes the relationship between two values that are not equal. Egality is not the same as inequality. Use the not equal symbol () in most cases when two values are not equal. To contrast values, whether they are tiny or huge, different inequalities are employed. By adjusting both sides until just the variables are left, you may eliminate many straightforward inequalities. However, several factors cause inequality: Divide or add negative values on both sides. Switch the left and right.

17[tex]\leq[/tex]2x - 3[tex]\leq[/tex]25

[tex]20\leq 2x\leq 28\\10\leq x\leq 14[/tex]

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Answer:

The statement "The value of f(–10) = 82" is not true. To find the value of f(–10), we substitute –10 for x in the equation:

Step-by-step explanation:

The statement "The value of f(–10) = 82" is not true. To find the value of f(–10), we substitute –10 for x in the equation:

f(–10) = (–10)^2 – 5(–10) + 12 = 100 – (–50) + 12 = 162

The statement "The graph of the function is a parabola" is true. Because the equation of the function is of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a is not equal to 0, the graph of the function is a parabola.

The statement "The graph of the function opens down." is true. The coefficient of x^2 is a positive value, which means the parabola opens down.

So, the true statements are

The graph of the function is a parabola.

The graph of the function opens down.

The graph does not contain the point (20, –8)

There are 749398 5-disc combinations that can be made from the collection.

Combination refers to the number of ways that a certain number of items can be selected from a larger set of items without regard to the order of the items. It is different from permutation, which takes into account the order of the items.

The number of combinations can be calculated using the formula:

nCr = n! / (r! x (n - r)!)

Where n is the total number of items in the collection, and r is the number of items in each combination

Since a movie collection has 6 mystery, 10 comedy, 4 adventure, 9 drama, and 12 science fiction discs, then the total number of discs in the collection is 41.

Plug in the values to the formula for combination.

nCr = n! / (r! x (n - r)!)

41C5 = 41! / (5! x (41 - 5)!)

41C5 = 41! / (5! x 36!)

41C5 = 749398

Therefore, there are 749398 5-disc combinations that can be made from the collection.

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By taking derivative, slope of the logarithmic function can be found.

The slope of a logarithmic function can be found by taking the derivative of the function with respect to x using the rules of logarithmic differentiation. The general form of the derivative of a logarithmic function is:

d/dx(log(f(x))) = (1/f(x))*(df/dx)

The slope means change in Y with respect to change in X. The slope is the ratio of the rise to run and it shows the steepness of the plane.

The slope is denoted by m.

Example:

Find the slope of y = log(x^2+1)

d/dx(log(x^2+1)) = (1/(x^2+1))*(2x) = 2x/(x^2+1)

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let y = number of child tickets so the total 0f 34 child tickets is sold at that day.

To calculate an expression's value by substitution, integers are substituted for letters in the expression.

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.

let x = number of adult tickets

let y = number of child tickets

x = 4y  (four times as many adult tickets as child tickets

8.4x + 5.1y = 1315.8

but we can substitute 4y for x

8.4(4y) + 5.1y = 1315.8

33.6y + 5.1y = 1315.8

38.7y = 1315.8

y = 34 and x = 4Y

x = 136

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Therefore , By dividing by the constant shown to the right, you can change any equation into intercept form:

=> x/3 + y/3 = 1 => x/1.2 +y/(-1) = 1

A mathematical equation is a formula that joins two statements with the equal sign, or =, to express equality. All mathematical formulas, including equations, have an equal sign. Algebra is frequently utilized in equations. Algebra is used in mathematics when we don't know the exact quantity to compute. A mathematical statement demonstrating the equality of two mathematical expressions is the definition of an equation in algebra. For instance, the variables 3x + 5 and 14 are separated by an equal sign in the equation 3x + 5 = 14.

Here,

Any equation can be changed to intercept form by dividing by the constant on the right:

=> x/3 + y/3 =1

=> x/1.2 + y/(-1) = 1

These inform you that only graph A is appropriate because the x- and y-intercepts of the first equation are 3 in this case. Given that it has an x-intercept of 1.2 and a y-intercept of -1, the second equation also matches with graph A.

Therefore , By dividing by the constant shown to the right, you can change any equation into intercept form:

=> x/3 + y/3 = 1 => x/1.2 +y/(-1) = 1

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Solutions refers to the answer to the problem, Real Solutions are realistic and practical whereas ideal solutions are the best cases for the solutions.

In mathematics, a solution refers to a value or set of values that satisfies a given equation, system of equations, or problem. For example, if the equation is x + 2 = 4, then the solution is x = 2, because substituting 2 for x in the equation makes it true. In the case of systems of equations or more complex problems, a solution may be a set of values that satisfies all the equations or conditions of the problem. In such case, the solution may be represented graphically or as coordinates of a point.

In mathematics, an ideal solution refers to a solution that meets all the desired criteria or requirements without any constraints. It is the "best case" scenario. A real solution, on the other hand, refers to a solution that takes into account all the limitations and constraints of a problem. It is a more practical and realistic solution.

Ideal Solutions - Best cases, Usually not possible.

Real Solutions - Practical and possible to attain.

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If Elliot purchased 3 trick cameras and 4 fake snakes for $40 and  Kaci bought 7 trick cameras and 8 fake snakes for $85 , then the system of equations that can be used to find the cost of trick camera and fake snakes are [tex]3x+4y=40[/tex] and [tex]7x+8y=85[/tex] .

Elliot purchased 3 trick cameras and 4 fake snakes for $40  ;

Kaci purchased 7 trick cameras and 8 fake snakes for  $85 ;

let the number of trick cameras purchased be = x ;

let the number of fake snakes purchased be = y ;

So , the equation for Elliot's purchase is ⇒ [tex]3x+4y=40[/tex] ;

and the equation for Kaci's purchase is ⇒ [tex]7x+8y=85[/tex] ;

Therefore , the equations to solve for the number of trick cameras and number of fake cameras are  [tex]3x+4y=40[/tex] and [tex]7x+8y=85[/tex] .

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Answer:2024

Step-by-step explanation:

In 8 months and 17 days Michael will reach the height of Caroline.

An equation is said to be linear if the maximum power of the variable is consistently 1. Another name for it is a one-degree equation. A linear equation with one variable has the conventional form Ax + B = 0. In this case, the variables x and A are variables, whereas B is a constant. A linear equation with two variables has the conventional form Ax + By = C. Here, the variables x and y, the coefficients A and B, and the constant C are all present.

A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.

Ax + B = 0, where A and B are both real integers, and x is the only variable, is the standard form or general form of a linear equation in one variable. Ax + By = C is the formula used to describe linear equations with two variables, where x and y are the variables and A, B, and C are any real values.

Let in x months Michael and Caroline will be at same height so,

36 + 0.75x = 39 + 0.4x

⇒0.35x = 3

⇒x = 8.57

So, in 8 months and .57*30 = 17 days Michael will reach the height of Caroline.

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When two lines intersect the vertically opposite angles so formed are equal in measure.

When two lines intersect each other, then the opposite angles, created due to the intersection are called vertical angles or vertically opposite angles.

We can see in the figure two lines intersect each other at a point. Therefore the two pairs of vertically opposite angles are ∠1 and ∠3, ∠2 and ∠4.

From the figure, ∠1 + ∠2 = 180°  [Linear pair]     eqn(1)

∠2 + ∠3 = 180°  [Linear pair]   eqn(2)

From eqn(1) and eqn(2), we get

∠1 + ∠2 = ∠2 + ∠3

∴ ∠1 = ∠3

Similarly, we can prove ∠2 = ∠4 also.

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The provided phrase, x3+4x2+ 7x+ â' 2 has four terms. and coefficient are 1 , 4 ,7 for the expression

What does the phrase mean?

Expressions are made up of terms. A term might be either a constant, a variable, or a combination of both.

What kind of phrase would that be?

A single letter, a single positive or negative integer, or a number of variables multiplied but never added or subtracted might be the only variable. A number is placed in front of some nouns that have variables. The number used before a sentence is called a coefficient. single-word illustrations: Three times in one phrase.

The mathematical operator + separates the four components in the equation x3+4x2+ 7x+â' 2.

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The y-intercept of the line y = 7x -38 is -38. And the x-intercept is 0

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.

Given:

The equation of the line,

y-4=7(x-6)

Simplify,

y -4 = 7x - 42

y = 7x -38

And slope-intercept form:

y = mx + b,

where m is the slope and y is the intercept.

Comparing with the slope intercept form,

we get,

m = 7 and b = -38

And the x-intercept is 0

Therefore3, y-intercept is -38.

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Answer:

The area of x is 4

The area of the right triangle is 6

Step-by-step explanation:

The Pythagorean Theorem can be used when dealing with right triangles. So, if we use the equation a^2 + b^2 = c^2, we can find that 3^2 + x^2 = 5^2 or 9 + x^2 = 25. If we calculate, we can find that x is equal to 4. Now if we multiply by 3 and divide by 2, we can get the answer of 6 as the area of the right triangle.

Therefore , the solution of the given problem of slope comes out to be

y = (-1/2)x + 8.

The slope-intercept formulation of an equation is used to represent each time the formula of a line is stated as y = mx + b. The line's slope is represented by the m. Additionally, b is the b in the y-intercepting point (0, b). For instance, the y-intercept is at (0, 7) and the slope is 3 in the formula y = 3x - 7.

Here,

Locate the slope first.

These two positions (0, 8) and are in the line, therefore we know that (6,5)

Slope: (y2-y1)/(x2 - x1)

Slope = 8-5 / 0-6

Slope = 3/-6 => Slope = -1/2

The gradient is -1/2. When x = 0, you can find the y-intercept. We also know that the line passes through the point (0,8), and since x = 0, the y-intercept is equal to 8.

using the slope-intercept form, which is

Using the equation y = mx + b, where m denotes the slope and b the y-intercept, we get:

y = (-1/2)x + 8

Therefore , the solution of the given problem of slope comes out to be

y = (-1/2)x + 8.

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Answer:

3/20

Step-by-step explanation:

To do this multiply the numerators (the top numbers) together first.

3 * 1 = 3

This is the numerator (the top number) of the answer fraction.

Next we take the bottom numbers:

5 * 4 = 20

We have 3/20 as our fraction. We know 3 is a prime number and can only be divided by 1 or 3. 20 cannot be divided by 1 or 3 so this fraction cannot be simplified any more.

The interquartile range (IQR) of the data in the dot plot is 3.5.

The interquartile range is a measure of where the “middle fifty” is in a data set. Where a range is a measure of where the beginning and end are in a set, an interquartile range is a measure of where the bulk of the values lie. That’s why it’s preferred over many other measures of spread when reporting things like school performance or SAT scores.

In descriptive statistics, the interquartile range is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, fourth spread, or H‑spread. It is defined as the difference between the 75th and 25th percentiles of the data.

The  upper and lower median values are shown below which would be the mean of the 2 middle values shown in the brackets below:

43, (44, 44,) 44, | 45, | 45, (47, 48,) 48

Upper median (Q3) = (47+48) ÷ 2 = 47.5

Lower median (Q1) = (44+44) ÷ 2 = 44

=>Step 4: Subtract the lower median from the upper median to get the IQR.

IQR = Upper median (Q3) - Lower median (Q1)

IQR = 47.5 - 44

IQR = 3.5

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Answer:

b.x=30

c.x=21

Step-by-step explanation:

[tex](\stackrel{x_1}{4}~,~\stackrel{y_1}{-6})\qquad (\stackrel{x_2}{-2}~,~\stackrel{y_2}{3}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{3}-\stackrel{y1}{(-6)}}}{\underset{\textit{\large run}} {\underset{x_2}{-2}-\underset{x_1}{4}}} \implies \cfrac{3 +6}{-6} \implies \cfrac{ 9 }{ -6 } \implies - \cfrac{3 }{ 2 }[/tex]

Answer:

9/-6 or -1.5

Step-by-step explanation:

Follow the steps in the image

-Hope this Helped

Answer: No Solutions

Step-by-step explanation:

8w + 5= 4 (2 w+1)=

8w+5=(4)(2w)+(4)(1)

8w+5=8w+4

    -8w      -8w

5=4

-5  -5

0=-1

Which means there are No Solutions.

Answer:

8w+5=4(2w+1)

8w+5=8w+4

collect like terms

8w-8w = 4-5

0=-1

there is no solution