Which is one of the transformations applied to the graph of $$ f(x)=x^{2} $$ to change it into the graph of $$ g(x)=4 x^{2}+24 x+ $$ The graph of $$ f(x)=x^{2} $$ is widened. The graph of $$ f(x)=x^{2} $$ is shifted left 3 units. The graph of $$ f(x)=x^{2} $$ is shifted up 30 units. The graph of $$ f(x)=x^{2} $$ is reflected over the $$ x $$-axis.

Question

UpStudy AI · Accepted Answer

To determine which transformation is applied to the graph of $$ f(x) = x^2 $$ to change it into the graph of $$ g(x) = 4x^2 + 24x + c $$, we first need to analyze the function $$ g(x) $$.

1. **Identify the form of $$ g(x) $$**:
   The function $$ g(x) = 4x^2 + 24x + c $$ is a quadratic function. We can rewrite it in vertex form by completing the square.

2. **Complete the square**:
   Start with the expression $$ 4x^2 + 24x $$:
   $$
   g(x) = 4(x^2 + 6x) + c
   $$
   Now, complete the square for $$ x^2 + 6x $$:
   $$
   x^2 + 6x = (x + 3)^2 - 9
   $$
   Substitute this back into $$ g(x) $$:
   $$
   g(x) = 4((x + 3)^2 - 9) + c
   $$
   $$
   g(x) = 4(x + 3)^2 - 36 + c
   $$
   $$
   g(x) = 4(x + 3)^2 + (c - 36)
   $$

3. **Analyze the transformations**:
   - The term $$ 4(x + 3)^2 $$ indicates that the graph of $$ f(x) = x^2 $$ is widened by a factor of 4 (since the coefficient of $$ x^2 $$ is 4).
   - The term $$ (x + 3) $$ indicates a horizontal shift to the left by 3 units.
   - The constant $$ c - 36 $$ indicates a vertical shift, but we need the specific value of $$ c $$ to determine how much it shifts up or down.

4. **Conclusion**:
   Based on the analysis, the transformations applied to $$ f(x) = x^2 $$ to obtain $$ g(x) = 4x^2 + 24x + c $$ include:
   - Widening the graph by a factor of 4.
   - Shifting the graph left by 3 units.

Since the question asks for one of the transformations, the correct answer is:

**The graph of $$ f(x) = x^2 $$ is widened.**
The graph of $$ f(x) = x^2 $$ is widened by a factor of 4.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions