Applying Transformations to the Parabola and Determining the Minimum or Maximum
Apply transformations to parabolas and find minimum or maximum values in Grade 10 algebra. Use vertex form f(x)=a(x-h)²+k to control shape, direction, and vertex position.
Key Concepts
New Concept The vertex form of a quadratic function is $f(x) = a(x h)^2 + k$.
Why it matters This vertex form isn't just about parabolas; it's your first look at a universal principle of transforming any function by adjusting simple parameters. Mastering this unlocks the ability to model and optimize real world scenarios, from the path of a projectile to the profits of a business.
What’s next Next, you’ll use this form to instantly identify a parabola's vertex, sketch its graph, and find its minimum or maximum value.
Common Questions
How does vertex form f(x) = a(x-h)² + k control a parabola?
The value a determines width and direction (narrow for |a|>1, wide for |a|<1, up for a>0, down for a<0). h shifts horizontally, k shifts vertically, and (h,k) is the vertex.
How do you find the minimum or maximum of a parabola?
The vertex (h, k) gives the minimum when a > 0 and the maximum when a < 0. The y-value k is the minimum or maximum value of the function.
How do transformations in vertex form affect the graph step by step?
Start with y = x², apply vertical stretch/compression by a, reflect over x-axis if a < 0, shift right h units (or left if h < 0), then shift up k units (or down if k < 0).