Quadratic Word Problems: Applications and Vertex

Quadratic Applications

• Projectile motion and Bridge arch problems can be modeled using quadratic equations
• Finding Maxima and Minima: e.g., Max area, height, etc. using quadratic equations

Revenue and Profit

• Revenue = Price × Quantity
• Profit = Revenue – Expenses = (# sold)(price) – (#sold)(cost price)
• Vertex represents maximum revenue or profit

Finding the Vertex of a Quadratic

• Completing the square to find the vertex
• Calculating the midpoint of the x-intercepts and then substituting back into equation to find y
• Using the formula x = -b / 2a to calculate the x-coordinate of the vertex (axis of symmetry) and then substituting back into equation to find y
• Vertex represents the highest/lowest point (max if a < 0 /min if a > 0)

Solving Quadratic Equations

• Factoring and applying the zero-product property to find x-intercepts
• Using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a to find x-intercepts
• X-intercepts represent where the object hits the ground (or similar)

Converting between Quadratic Forms

• Converting between standard form, vertex form, and factored form
• Refer to Flowchart on Page 3 for summary

Graphing Quadratic Functions

• Graphing quadratic functions in standard form: y = ax² + bx + c
• Graphing quadratic functions in vertex form: y = a(x – h)² + k, where (h, k) = vertex
• Graphing quadratic functions in factored form: y = a(x – r)(x – s), where r, s are the x-intercepts
• Key points to label: y-intercept, x-intercepts, vertex

Domain and Range

• Determining the Domain and Range for quadratic functions using proper notation