Example Card: Finding Maximum Height Using the Axis of Symmetry
Find maximum height of a quadratic function in Grade 9 algebra using the axis of symmetry x=-b/(2a) to locate the vertex, then substitute to calculate the maximum y-value.
Key Concepts
This formula looks complex, but it holds the secret to a rocket's peak flight. Let's find it using the key idea of applying the axis of symmetry formula.
Example Problem A model rocket's height in feet is given by $y = 16t^2 + 192t + 20$. Find its maximum height and how long it takes to reach it.
Step by step 1. The maximum height occurs at the vertex of the parabola. We can find the time ($t$) it takes to reach the vertex using the formula for the axis of symmetry, $t = \frac{b}{2a}$. 2. In the equation $y = 16t^2 + 192t + 20$, we have $a = 16$ and $b = 192$. Substitute these values into the formula: $$t = \frac{192}{2( 16)}$$ 3. Simplify the expression to find the time: $$t = \frac{192}{ 32} = 6$$ 4. Now, substitute $t = 6$ back into the original equation to find the maximum height ($y$): $$y = 16(6)^2 + 192(6) + 20$$ 5. Simplify the equation to find the height: $$y = 16(36) + 1152 + 20$$ $$y = 576 + 1152 + 20 = 596$$ 6. The vertex is at $(6, 596)$. This means the rocket reaches a maximum height of 596 feet after 6 seconds.
Common Questions
What is the axis of symmetry of a parabola?
The axis of symmetry is a vertical line that passes through the vertex of a parabola, dividing it into two mirror-image halves. For a quadratic y = ax² + bx + c, it is the vertical line x = -b/(2a).
How do you use the axis of symmetry to find maximum height?
First compute x = -b/(2a) to find the x-coordinate of the vertex. Then substitute that x-value back into the original quadratic equation to find the maximum y-value (the height at the vertex).
When does a parabola have a maximum versus a minimum?
A parabola opens downward (a < 0) and has a maximum value at its vertex. A parabola opens upward (a > 0) and has a minimum value at its vertex. For projectile/height problems, a is typically negative.