^x+1 +1 describe transformation","","Solution:\u003Cbr />\n1. Given function:\u003Cbr />\n * \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = -2$4$^{x+1} + 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Base function:\u003Cbr />\n * \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = 4^x\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Identify transformations step-by-step:\u003Cbr />\n - **Translation horizontally**: The function has \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$x+1$\u003C/math-field>\u003C/math-field> as the exponent instead of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>. This indicates a horizontal shift to the left by 1 unit.\u003Cbr />\n - **Vertical stretch and reflection**: The coefficient before \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-2\u003C/math-field>\u003C/math-field>.\u003Cbr />\n - **Vertical stretch**: The factor \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\u003C/math-field>\u003C/math-field> indicates that the function is stretched vertically by a factor of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\u003C/math-field>\u003C/math-field>.\u003Cbr />\n - **Reflection**: The negative sign indicates a reflection across the x-axis.\u003Cbr />\n - **Vertical translation**: The \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>+1\u003C/math-field>\u003C/math-field> outside the function indicates a vertical shift upwards by 1 unit.\u003Cbr />\n\u003Cbr />\n4. Describe the complete transformation:\u003Cbr />\n - The function \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = 4^x\u003C/math-field>\u003C/math-field> undergoes the following transformations: a horizontal shift to the left by 1 unit, a vertical stretch by a factor of 2, reflection across the x-axis, and finally a vertical shift upwards by 1 unit.",1255,251,"y-2-4-x-1-1-describe-transformation",{"id":55,"category":47,"text_question":56,"photo_question":49,"text_answer":57,"step_text_answer":8,"step_photo_answer":8,"views":58,"likes":59,"slug":60},538072,"Andrew's age is three times John's plus nine years, and their ages add up to 69 years. Find both ages.","Solution:\u003Cbr />\n1. Define variables:\u003Cbr />\n- Let \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> be Juan's age.\u003Cbr />\n- Andrés' age is then \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x + 9\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n2. Set up the equation for the total age:\u003Cbr />\n- Juan's age \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> plus Andrés' age \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x + 9\u003C/math-field>\u003C/math-field> equals 69.\u003Cbr />\n\u003Cbr />\n3. Equation:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x + $3x + 9$ = 69\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Simplify and solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x + 3x + 9 = 69\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x + 9 = 69\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x = 60\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 15\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Find Andrés' age:\u003Cbr />\n- Substitute \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 15\u003C/math-field>\u003C/math-field> into Andrés' age expression:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x + 9 = 3$15$ + 9 = 45 + 9 = 54\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Therefore, the ages are:\u003Cbr />\n- Juan: 15 years\u003Cbr />\n- Andrés: 54 years",531,106,"andrew-s-age-is-three-times-john-s-plus-nine-years-and-their-ages-add-up-to-69-years-find-both-ages",{"id":62,"category":47,"text_question":63,"photo_question":49,"text_answer":64,"step_text_answer":8,"step_photo_answer":8,"views":65,"likes":66,"slug":67},538062,"5/8 *18","1. Multiply the numerator of the fraction by the whole number: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 5 \\times 18 = 90 \u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n2. Divide the result by the denominator of the fraction:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{90}{8} = 11.25 \u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\nTherefore, the final answer is 11.25.",938,188,"5-8-18",{"id":69,"category":47,"text_question":70,"photo_question":49,"text_answer":71,"step_text_answer":8,"step_photo_answer":8,"views":72,"likes":73,"slug":74},537955,"How much water should be added to 1 gallon of pure antifreeze to obtain the solution that is 45% antifreeze","Step 1: Let \\( x \\) be the amount of water to add.\u003Cbr />\n\u003Cbr />\nStep 2: The total volume of the solution will be \\( 1 + x \\) gallons.\u003Cbr />\n\u003Cbr />\nStep 3: The concentration of antifreeze in the solution is given by:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{1 \\, \\text{gallon of antifreeze}}{1 + x} = 0.45 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nStep 4: Solving for \\( x \\):\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 1 = 0.45(1 + x) \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 1 = 0.45 + 0.45x \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 1 - 0.45 = 0.45x \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 0.55 = 0.45x \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = \\frac{0.55}{0.45} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x \\approx 1.222 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nAnswer: Approximately 1.222 gallons of water should be added.",1220,244,"how-much-water-should-be-added-to-1-gallon-of-pure-antifreeze-to-obtain-the-solution-that-is-45-antifreeze",{"id":76,"category":47,"text_question":77,"photo_question":49,"text_answer":78,"step_text_answer":8,"step_photo_answer":8,"views":65,"likes":66,"slug":79},537945,"The area of Ms. McCarthy's garden measures 600 square feet. She plants vegetables in 75% of her garden and flowers in the remaining section.\nWhat is the area of the garden that Ms. McCarthy plants with flowers?","Calculate the area of the garden that is planted with flowers: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>100\\% - 75\\% = 25\\%\u003C/math-field>\u003C/math-field> of the garden is for flowers. Therefore, the area planted with flowers is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.25 \\times 600 = 150 \\text{ square feet}\u003C/math-field>\u003C/math-field>. Answer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>150 \\text{ square feet}\u003C/math-field>\u003C/math-field>.","the-area-of-ms-mccarthy-s-garden-measures-600-square-feet-she-plants-vegetables-in-75-of-her-garden-and-flowers-in-the-remaining-section-what-is-the-area-of-the-garden-that-ms-mccarthy-plants-wit",{"id":81,"category":47,"text_question":82,"photo_question":49,"text_answer":83,"step_text_answer":8,"step_photo_answer":8,"views":84,"likes":85,"slug":86},537936,"There is a 6% probability that a randomly selected employed person has more than one job. There is 40% probability that a randomly selected employed person is male, given that the person has more than one job. What is the probability that a randomly selected employee is a man with more than one job? What is the probability that a randomly selected employee is a man and has more than one job?","1. Identify the probability that a person has more than one job: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P(M) = 0.06\u003C/math-field>\u003C/math-field>\u003Cbr />\n2. Identify the probability that a person is a male given they have more than one job: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P(A|M) = 0.40\u003C/math-field>\u003C/math-field>\u003Cbr />\n3. Use the conditional probability formula: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P(A \\cap M) = P(A|M) \\times P(M)\u003C/math-field>\u003C/math-field>\u003Cbr />\n4. Calculate: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.40 \\times 0.06 = 0.024\u003C/math-field>\u003C/math-field>\u003Cbr />\n5. Therefore, the probability that a randomly selected employee is a man with more than one job is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.024\u003C/math-field>\u003C/math-field>.",1114,223,"there-is-a-6-probability-that-a-randomly-selected-employed-person-has-more-than-one-job-there-is-40-probability-that-a-randomly-selected-employed-person-is-male-given-that-the-person-has-more-tha",{"id":88,"category":47,"text_question":89,"photo_question":49,"text_answer":90,"step_text_answer":8,"step_photo_answer":8,"views":91,"likes":92,"slug":93},537897,"Is 4 a factor of 1,671,349? \nHow do you know without multiplying or dividing?","1. Identify the last two digits of the number 1,671,349: 49.\u003Cbr />\n2. Check if 49 is divisible by 4:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 49 \\div 4 = 12.25 \u003C/math-field>\u003C/math-field>\u003Cbr />\n Since 12.25 is not an integer, 49 is not divisible by 4.\u003Cbr />\n3. Conclude that 1,671,349 is not divisible by 4.\u003Cbr />\n\u003Cbr />\nAnswer: 4 is not a factor of 1,671,349.",1093,219,"is-4-a-factor-of-1-671-349-how-do-you-know-without-multiplying-or-dividing",{"id":95,"category":47,"text_question":96,"photo_question":49,"text_answer":97,"step_text_answer":8,"step_photo_answer":8,"views":98,"likes":99,"slug":100},537867,"If you buy a car where you give a down payment of $ 998 and the payment will be $ 465 monthly\n for 5 ½ years at an interest rate of 8.5% per annum computed monthly. What was the\n sale price?","Solution:\u003Cbr>\u003Cbr>1. Given:\u003Cbr>\u003Cbr>- Down payment: USD 998\u003Cbr>\u003Cbr>- Monthly payment: USD 465\u003Cbr>\u003Cbr>- Loan term: 5.5 years or 66 months\u003Cbr>\u003Cbr>- Annual interest rate: 8.5% (monthly rate \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= \\frac{8.5}{12}\u003C/math-field>\u003C/math-field> percent = 0.70833% = 0.0070833)\u003Cbr>\u003Cbr>2. Use the formula for the present value of an annuity to find the loan amount:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Loan Amount} = P \\times \\frac{1 - (1 + r)^{-n}}{r}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>where\u003Cbr>\u003Cbr>- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P\u003C/math-field>\u003C/math-field> is the monthly payment (USD 465)\u003Cbr>\u003Cbr>- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>r\u003C/math-field>\u003C/math-field> is the monthly interest rate (0.0070833)\u003Cbr>\u003Cbr>- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>n\u003C/math-field>\u003C/math-field> is the total number of payments (66)\u003Cbr>\u003Cbr>3. Substitute and compute:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Loan Amount} = 465 \\times \\frac{1 - (1 + 0.0070833)^{-66}}{0.0070833}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Calculate the value:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Loan Amount}\\approx465\\times50.9965\\approx24446.96\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>5. Add the down payment to get the total sale price:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Sale Price} = \\text{Loan Amount} + \\text{Down Payment}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Sale Price}=24446.96+998\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Sale Price}=25444.96\u003C/math-field>\u003C/math-field> USD\u003Cdiv>\u003Cbr>\u003Cbr>\u003C/div>\u003Cdiv>\u003Cbr>\u003Cbr>\u003C/div>\u003Cdiv>\u003Cbr>\u003Cbr>\u003C/div>\u003Cdiv>\u003Cbr>\u003Cbr>\u003C/div>\u003Cdiv>\u003Cbr>\u003Cbr>\u003C/div>",343,69,"if-you-buy-a-car-where-you-give-a-down-payment-of-998-and-the-payment-will-be-465-monthly-for-5-years-at-an-interest-rate-of-8-5-per-annum-computed-monthly-what-was-the-sale-price",{"id":102,"category":47,"text_question":103,"photo_question":49,"text_answer":104,"step_text_answer":8,"step_photo_answer":8,"views":105,"likes":106,"slug":107},537853,"60 percent of 27000","\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Solution:}\u003C/math-field>\u003C/math-field> \u003Cbr />\n1. To find 60 percent of 27000, multiply 27000 by \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{60}{100}\u003C/math-field>\u003C/math-field>: \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>27000 \\times \\frac{60}{100}\u003C/math-field>\u003C/math-field>. \u003Cbr />\n2. Simplify the fraction by reducing it: \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{60}{100} = \\frac{3}{5}\u003C/math-field>\u003C/math-field>. \u003Cbr />\n3. Multiply 27000 by \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{3}{5}\u003C/math-field>\u003C/math-field>: \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>27000 \\times \\frac{3}{5} = \\frac{27000 \\times 3}{5}\u003C/math-field>\u003C/math-field>. \u003Cbr />\n4. Calculate \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>27000 \\times 3\u003C/math-field>\u003C/math-field>: \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>27000 \\times 3 = 81000\u003C/math-field>\u003C/math-field>. \u003Cbr />\n5. Divide 81000 by 5: \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{81000}{5} = 16200\u003C/math-field>\u003C/math-field>. \u003Cbr />\n6. Thus, 60 percent of 27000 is 16200.",799,160,"60-percent-of-27000",{"id":109,"category":47,"text_question":110,"photo_question":49,"text_answer":111,"step_text_answer":8,"step_photo_answer":8,"views":112,"likes":113,"slug":114},537835,"The temperature fell below zero degrees Fahrenheit on 49 days out of the 91 days of winter. For what percent of winter days did the temperature stay above zero degrees Fahrenheit. Round to the nearest percent","1. Total winter days: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>91\u003C/math-field>\u003C/math-field>\u003Cbr />\n2. Days below zero: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>49\u003C/math-field>\u003C/math-field>\u003Cbr />\n3. Days above zero: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>91 - 49 = 42\u003C/math-field>\u003C/math-field>\u003Cbr />\n4. Percentage of days above zero: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left(\\frac{42}{91}\\right) \\times 100 \\approx 46.1538\u003C/math-field>\u003C/math-field>\u003Cbr />\n5. Rounded percent: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>46\\%\u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cbr />\nAnswer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field 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