Question

Question 1
1 pts
Use Desmos to identify key features of the function y = 4x²+4x+1.
Select all that are true.
The y-intercept is at (0,4)
The function has 1 x-intercept
The equation of the axis of symmetry of the graph of the function is 2 = 2
The y-intercept is at (0,1)
The maximum value of the function is x = 2
The function has 2 zeros
The minimum value of the function is 0

Answer 1

To analyze the function \( y = 4x^2 + 4x + 1 \), we can identify key features step by step.

1. **Y-intercept**: To find the y-intercept, substitute \( x = 0 \) into the equation:
\[
y = 4(0)^2 + 4(0) + 1 = 1
\]
So, the y-intercept is at \( (0, 1) \).

2. **X-intercepts**: To find the x-intercepts, set \( y = 0 \) and solve for \( x \):
\[
4x^2 + 4x + 1 = 0
\]
We can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 4, b = 4, c = 1 \):
\[
x = \frac{-4 \pm \sqrt{(4)^2 - 4(4)(1)}}{2(4)} = \frac{-4 \pm \sqrt{16 - 16}}{8} = \frac{-4}{8} = -\frac{1}{2}
\]
Since the discriminant (b² - 4ac) is 0, there is 1 x-intercept at \( x = -\frac{1}{2} \).

3. **Axis of symmetry**: The axis of symmetry for a quadratic \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \):
\[
x = -\frac{4}{2 \cdot 4} = -\frac{1}{2}
\]
Therefore, the correct equation for the axis of symmetry is \( x = -\frac{1}{2} \), not \( 2 = 2 \).

4. **Maximum/Minimum Value**: Since the coefficient of \( x^2 \) (which is 4) is positive, the parabola opens upwards, indicating a minimum value. To find the minimum \( y \) value, substitute \( x = -\frac{1}{2} \) back into the original equation:
\[
y = 4\left(-\frac{1}{2}\right)^2 + 4\left(-\frac{1}{2}\right) + 1 = 4\left(\frac{1}{4}\right) - 2 + 1 = 1 - 2 + 1 = 0
\]
So, the minimum value of the function is 0.

Now, let's summarize the statements:

- The y-intercept is at \((0, 4)\) **(False)**
- The function has 1 x-intercept **(True)**
- The equation of the axis of symmetry of the graph of the function is \( 2 = 2 \) **(False)**
- The y-intercept is at \((0, 1)\) **(True)**
- The maximum value of the function is \( x = 2 \) **(False)**
- The function has 2 zeros **(False)**
- The minimum value of the function is 0 **(True)**

Thus, the true statements are:
- The function has 1 x-intercept.
- The y-intercept is at (0, 1).
- The minimum value of the function is 0.