Question

Find the​ y-intercept, the axis of​ symmetry, and the vertex of the graph of the function.
​f(x)equals5 x squared plus 5 x plus 15
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Part 1
The​ y-intercept is
  
enter your response here. ​(Type an ordered​ pair.)

Answer 1

To find the \( y \)-intercept of the function \( f(x) = 5x^2 + 5x + 15 \), we need to evaluate \( f(x) \) at \( x = 0 \).

Calculating:
\[
f(0) = 5(0)^2 + 5(0) + 15 = 15
\]

Therefore, the \( y \)-intercept is the ordered pair \( (0, 15) \).

### Part 1 Answer:
The \( y \)-intercept is \( (0, 15) \).

### Part 2: Finding the Axis of Symmetry
The axis of symmetry for a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
In this case, \( a = 5 \) and \( b = 5 \).

Calculating:
\[
x = -\frac{5}{2 \times 5} = -\frac{5}{10} = -\frac{1}{2}
\]

### Part 2 Answer:
The axis of symmetry is \( x = -\frac{1}{2} \).

### Part 3: Finding the Vertex
To find the vertex, we calculate the \( y \)-coordinate using \( f(x) \) at the axis of symmetry \( x = -\frac{1}{2} \).

Calculating:
\[
f\left(-\frac{1}{2}\right) = 5\left(-\frac{1}{2}\right)^2 + 5\left(-\frac{1}{2}\right) + 15
\]
\[
= 5\left(\frac{1}{4}\right) - \frac{5}{2} + 15
\]
\[
= \frac{5}{4} - \frac{10}{4} + \frac{60}{4}
\]
\[
= \frac{5 - 10 + 60}{4} = \frac{55}{4}
\]

So the vertex is:
\[
\left(-\frac{1}{2}, \frac{55}{4}\right)
\]

### Part 3 Answer:
The vertex is \( \left(-\frac{1}{2}, \frac{55}{4}\right) \).

### Summary:
- **Y-intercept**: \( (0, 15) \)
- **Axis of symmetry**: \( x = -\frac{1}{2} \)
- **Vertex**: \( \left(-\frac{1}{2}, \frac{55}{4}\right) \)