- For the equation \(y = 2x^2 + 12x - 4\), the axis of symmetry is \(x = -3\) and the vertex is \((-3, -22)\).
- For the equation \(y = x^2 - 8x + 8\), the vertex form obtained by completing the square is \(y = (x - 4)^2 - 8\).
- Final results: \(\text{Vertex: } (-3, -22), \text{Axis of symmetry: } x = -3\); \(\text{Vertex form: } y = (x - 4)^2 - 8\).
Explanation
- Analysis of the Problem
We are given two quadratic equations: \(y = 2x^2 + 12x - 4\) and \(y = x^2 - 8x + 8\). Our objectives are to identify the vertex and axis of symmetry for the first equation, and to convert the second equation into vertex form \(y = a(x - h)^2 + k\). We recall that for a quadratic equation in standard form \(y = ax^2 + bx + c\), the axis of symmetry is given by \(x = -\frac{b}{2a}\) and the vertex \((h, k)\) is given by \(h = -\frac{b}{2a}\) and \(k = f(h)\). Completing the square is a standard algebraic method to convert standard form into vertex form.
- Vertex and Axis of Symmetry for the First Equation
For the equation \(y = 2x^2 + 12x - 4\), we identify the coefficients \(a = 2\), \(b = 12\), and \(c = -4\).
The axis of symmetry is calculated as:
\[x = -\frac{b}{2a} = -\frac{12}{2 \times 2} = -\frac{12}{4} = -3.\]
Now, we find the \(y\)-coordinate of the vertex by evaluating the function at \(x = -3\):
\[y = 2(-3)^2 + 12(-3) - 4 = 2(9) - 36 - 4 = 18 - 36 - 4 = -22.\]
Therefore, the vertex is \((-3, -22)\) and the axis of symmetry is \(x = -3\). This matches option (a/c).
- Converting the Second Equation to Vertex Form
For the equation \(y = x^2 - 8x + 8\), we want to write it in the form \(y = a(x - h)^2 + k\). We start by grouping the \(x\) terms:
\[y = (x^2 - 8x) + 8.\]
To complete the square for the quadratic expression \(x^2 - 8x\), we take half of the coefficient of \(x\), which is \(\frac{-8}{2} = -4\), and square it to get \((-4)^2 = 16\). We add and subtract 16 within the expression:
\[y = (x^2 - 8x + 16 - 16) + 8.\]
\[y = (x^2 - 8x + 16) - 16 + 8.\]
Recognizing \(x^2 - 8x + 16\) as \((x - 4)^2\), we simplify:
\[y = (x - 4)^2 - 8.\]
Thus, the vertex form is \(y = (x - 4)^2 - 8\), which matches option (c).
- Final Conclusion
We have identified the vertex and axis of symmetry for \(y = 2x^2 + 12x - 4\) as \((-3, -22)\) and \(x = -3\). We also successfully converted \(y = x^2 - 8x + 8\) to vertex form as \(y = (x - 4)^2 - 8\). Final answers are confirmed.
Examples
In physics, quadratic functions often model the trajectory of a projectile. If a ball is thrown into the air, its height \(h\) at time \(t\) follows a path like \(h(t) = at^2 + bt + c\). Finding the vertex of this parabola allows us to determine the maximum height the ball reaches and the time at which it occurs. Understanding how to find the vertex and write a quadratic equation in vertex form is essential for analyzing motion and optimizing physical systems.