Question

Use the vertex and intercepts to sketch the graph of the
quadratic function. Give the equation of the parabola's
axis of symmetry. Use the graph to determine
the function's domain and range. Use the graphing tool to graph the equation. Use the
vertex and one of the intercepts to graph the equation.
f(x) =x^2+5x-6
The axis of symmetry is blank
(Type an equation.)

Identify the function's domain.
The domain is blank
(Type the answer in interval notation.)

Identify the function's range.
The range is blank
(Type the answer in interval notation.)

Answer 1

To analyze the quadratic function \( f(x) = x^2 + 5x - 6 \), we can first find the vertex and intercepts.

### Step 1: Find the Vertex
The vertex of a parabola represented by \( f(x) = ax^2 + bx + c \) can be found using the formula:

\[
x = -\frac{b}{2a}
\]

In this case, \( a = 1 \) and \( b = 5 \):

\[
x = -\frac{5}{2 \cdot 1} = -\frac{5}{2} = -2.5
\]

Now, we find the y-coordinate of the vertex by plugging \( x = -2.5 \) back into the function:

\[
f(-2.5) = (-2.5)^2 + 5(-2.5) - 6
\]
\[
= 6.25 - 12.5 - 6
\]
\[
= 6.25 - 18.5 = -12.25
\]

So, the vertex is \( \left(-2.5, -12.25\right) \).

### Step 2: Find the Intercepts
**Y-intercept:**
To find the y-intercept, set \( x = 0 \):

\[
f(0) = 0^2 + 5(0) - 6 = -6
\]

So, the y-intercept is \( (0, -6) \).

**X-intercepts:**
To find the x-intercepts, set \( f(x) = 0 \):

\[
x^2 + 5x - 6 = 0
\]

We can factor this quadratic:

\[
(x + 6)(x - 1) = 0
\]

So, the x-intercepts are \( x + 6 = 0 \) (i.e., \( x = -6 \)) and \( x - 1 = 0 \) (i.e., \( x = 1 \)). Thus, the x-intercepts are \( (-6, 0) \) and \( (1, 0) \).

### Step 3: Axis of Symmetry
The axis of symmetry of the parabola can be found using the x-coordinate of the vertex:

\[
x = -2.5
\]

**The axis of symmetry is:**
\[
x = -\frac{5}{2}
\]

### Step 4: Domain and Range
**Domain:**
The domain of a quadratic function is always all real numbers, so in interval notation:

\[
\text{Domain: } (-\infty, \infty)
\]

**Range:**
Since this parabola opens upwards (as \( a > 0 \)), the range starts from the y-coordinate of the vertex:

\[
\text{Range: } [-12.25, \infty)
\]

### Summary
- **The axis of symmetry is:** \( x = -\frac{5}{2} \)
- **The domain is:** \( (-\infty, \infty) \)
- **The range is:** \( [-12.25, \infty) \)

You can now use this information to sketch the graph.