Write the quadratic function in the form $$ y=a(x-h)^{2}+k $$ and sketch its graph. $$ y=x^{2}+12 x $$

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UpStudy AI · Accepted Answer

Function by following steps:
- step0: Rewrite in vertex form:
  $$y=x^{2}+12x$$
- step1: Group the terms:
  $$y=\left(x^{2}+12x\right)$$
- step2: Add and subtract:
  $$y=\left(x^{2}+12x+36-36\right)$$
- step3: Use the distributive property:
  $$y=\left(x^{2}+12x+36\right)-36\times 1$$
- step4: Factor the expression:
  $$y=\left(x+6\right)^{2}-36\times 1$$
- step5: Calculate:
  $$y=\left(x+6\right)^{2}-36$$
To rewrite the quadratic function $$ y = x^2 + 12x $$ in the vertex form $$ y = a(x - h)^2 + k $$, we can complete the square.

1. **Identify the coefficients**: The given function is $$ y = x^2 + 12x $$. Here, $$ a = 1 $$ and $$ b = 12 $$.

2. **Complete the square**:
   - Take half of the coefficient of $$ x $$ (which is 12), square it, and add and subtract it inside the equation:
   $$
   y = x^2 + 12x = (x^2 + 12x + 36) - 36
   $$
   - This simplifies to:
   $$
   y = (x + 6)^2 - 36
   $$

3. **Vertex form**: Now we have the function in vertex form:
   $$
   y = (x + 6)^2 - 36
   $$
   Here, $$ h = -6 $$ and $$ k = -36 $$.

### Summary
The quadratic function in vertex form is:
$$
y = (x + 6)^2 - 36
$$

### Graph Sketch
While I cannot plot the graph directly, I can describe it:
- The vertex of the parabola is at the point $$(-6, -36)$$.
- The parabola opens upwards (since $$ a = 1 > 0 $$).
- The axis of symmetry is the line $$ x = -6 $$.

You can sketch the graph by plotting the vertex and a few additional points, such as:
- When $$ x = -5 $$, $$ y = (-5 + 6)^2 - 36 = 1 - 36 = -35 $$.
- When $$ x = -7 $$, $$ y = (-7 + 6)^2 - 36 = 1 - 36 = -35 $$.

This will give you a U-shaped curve centered at the vertex.
The quadratic function $$ y = x^2 + 12x $$ in vertex form is $$ y = (x + 6)^2 - 36 $$. The graph is a parabola opening upwards with its vertex at $$(-6, -36)$$.

Write the quadratic function in the form \( y=a(x-h)^{2}+k \) and sketch its graph. \( y=x^{2}+12 x \)

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