Algebra 3.7 Formulas and Functions Formulas A formula is an algebraic equation that relates two or more real-life quantities Formulas have more than one variable Formulas are used in science, business, geometry and everyday life. rt = d Distance Formula p = 2l + 2w Perimeter of a rectangle A = ½bh Area of a triangle V = lwh Volume of a rectangular prism C = 5/9 (F-32) Temperature conversion (F to C) Solving a formula When we solve a formula for an indicated variable, you simply isolate that variable on one side of the equation. To do this, you use inverse operations. Solve the triangle area formula: Solve for b A = ½bh A = ½bh Ask yourself, where is the b and what’s happening to it. (2) A = ½bh (2) The b is being multiplied by ½ and by h, so you want to divide by½ (multiply by 2)….. 2A = bh 2A = bh h h 2A = b h and divide by h. So, b = 2A h Solve the rectangle perimeter formula: Solve for w P = 2l + 2w P = 2l + 2w Ask yourself, where is the w and what’s happening to it. P -2l The w is being multiplied by 2 and that product is being added to 2l, so first subtract 2l….. = 2l + 2w -2l P – 2l = 2w P – 2l = 2w 2 2 P – 2l = w 2 and then divide by 2. So, w = P – 2l 2 Try these yourself! Solve for h: Answer: V = l wh h= V lw Solve for F: C = 5 (F – 32) 9 Hint: First isolate (F – 32) Answer: F = 9 C + 32 5 Functions A function is a rule that establishes a relationship between two quantities, called the input and the output A linear function usually uses the variable x to describe input, and y to describe output A two-variable equation is written in function form if the y is isolated on one side of the equation Function form: The output y y = 2x + 4 is a function of the input x Writing Function Form (you will learn why you do this later) When you write the equation in function form, arrange the terms in the order below: Function form: Put the y on the left y = mx + b Put the x term first on the right Put the constant last Write the fractions out as individual terms, such as 4 x and 5 1x 3 Rewrite the equation in Function Form (isolate y) 2x + 3y = 10 2x + 3y = 10 Ask yourself, where is the y and what’s happening to it. 2x + 3y = 10 -2x -2x The y is being multiplied by 3 and that product is being added to 2x, so first subtract 2x….. 3y = -2x + 10 3y = -2x + 10 3 3 y = - 2x + 10 3 3 and divide by 3. So, y = - 2x + 10 3 3 Rewrite the equation in Function Form (isolate y) y - 7 = 7x - 9x 5 y - 7 = -2x 5 +7 +7 (5) y = (-2x + 7) (5) 5 y = -10x + 35 First, simplify the right side. Next, add 7 to get the y term alone Then, multiply by 5. Make sure that you multiply EVERY TERM on BOTH SIDES by 5 So, y = - 10x + 35 Now you try these Rewrite in function form: x + 2y = 6 Answer: y = -1/2x + 3 Rewrite in function form: 1 y + 3 = -5x 4 Answer: y = -20x - 12 Homework pg. 177 # 11-29, 34-39 (calculator OK on these)