Question

quadratic equations graphs are parabolas. standard form: y = ax²+bx + c. vertex form: f(x)=a(x - h)² + k. vertex = highest or lowest point on a parabola. vertex x - value x=-b/2a, plug in to find y - value. *vertex (h,k). g(x)=a(x - h)² + k. how do the constants a, h, and k affect the graph of the quadratic function g(x)=a(x - h)² + k? write the equation of the quadratic function whose graph is shown at the right. explain your reasoning. then use technology to check your answer. g(x)=(x - 1)²

Explanation:

Step1: Analyze the effect of \(a\)

The value of \(a\) in \(g(x)=a(x - h)^2 + k\) determines the direction and width of the parabola. If \(a>0\), the parabola opens upward; if \(a < 0\), it opens downward. The magnitude of \(a\) affects the width. A larger \(|a|\) makes the parabola narrower and a smaller \(|a|\) makes it wider.

Step2: Analyze the effect of \(h\)

The value of \(h\) in \(g(x)=a(x - h)^2 + k\) is the \(x\) - coordinate of the vertex. It shifts the parabola horizontally. If \(h>0\), the parabola shifts \(h\) units to the right compared to \(y = ax^2\), and if \(h < 0\), it shifts \(|h|\) units to the left.

Step3: Analyze the effect of \(k\)

The value of \(k\) in \(g(x)=a(x - h)^2 + k\) is the \(y\) - coordinate of the vertex. It shifts the parabola vertically. If \(k>0\), the parabola shifts \(k\) units up compared to \(y = ax^2\), and if \(k < 0\), it shifts \(|k|\) units down.

Answer:

The coefficient \(a\) affects the direction (opens up if \(a>0\), down if \(a < 0\)) and width of the parabola. The value \(h\) shifts the parabola horizontally and \(k\) shifts it vertically.