Find the vertex, axis of symmetry, x-intercepts, y-intercept, value of the max/min, domain, and range of the following quadratics and then graph the parabola.
1. f(x) = 3x^2
vertex
x-int
max/min
domain
axis
y-int
value
range
2. f(x) = X^2 + 2x + 1
vertex
x-int
max/min
domain
axis
y-int
value
range

Let's break down and solve the quadratic functions one by one:

### 1. \( f(x) = 3x^2 \)

#### Vertex:
For a general quadratic function \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
In this case, \( a = 3 \), \( b = 0 \), and \( c = 0 \), so:
\[
x = -\frac{0}{2(3)} = 0
\]
The y-coordinate of the vertex is found by substituting \( x = 0 \) into the function:
\[
f(0) = 3(0)^2 = 0
\]
Thus, the vertex is at \( (0, 0) \).

#### Axis of Symmetry:
The axis of symmetry is the vertical line that passes through the vertex. Since the vertex is at \( x = 0 \), the axis of symmetry is:
\[
x = 0
\]

#### X-Intercepts:
To find the x-intercepts, we set \( f(x) = 0 \):
\[
3x^2 = 0 \quad \Rightarrow \quad x = 0
\]
There is only one x-intercept at \( (0, 0) \).

#### Y-Intercept:
The y-intercept is the value of \( f(x) \) when \( x = 0 \):
\[
f(0) = 0
\]
Thus, the y-intercept is also at \( (0, 0) \).

#### Maximum/Minimum Value:
Since the coefficient of \( x^2 \) is positive (\( a = 3 \)), the parabola opens upwards and has a **minimum value** at the vertex. The minimum value of \( f(x) \) is:
\[
f(0) = 0
\]

#### Domain:
The domain of any quadratic function is all real numbers:
\[
\text{Domain: } (-\infty, \infty)
\]

#### Range:
Since the parabola opens upwards and the minimum value is \( 0 \), the range is:
\[
\text{Range: } [0, \infty)
\]

---

### 2. \( f(x) = x^2 + 2x + 1 \)

#### Vertex:
Using the same vertex formula, \( a = 1 \), \( b = 2 \), and \( c = 1 \), so:
\[
x = -\frac{2}{2(1)} = -1
\]
Substituting \( x = -1 \) into the function:
\[
f(-1) = (-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0
\]
Thus, the vertex is at \( (-1, 0) \).

#### Axis of Symmetry:
The axis of symmetry is the vertical line that passes through the vertex. Since the vertex is at \( x = -1 \), the axis of symmetry is:
\[
x = -1
\]

#### X-Intercepts:
To find the x-intercepts, set \( f(x) = 0 \):
\[
x^2 + 2x + 1 = 0
\]
Factoring gives:
\[
(x + 1)^2 = 0
\]
Thus, \( x = -1 \), so there is one x-intercept at \( (-1, 0) \).

#### Y-Intercept:
The y-intercept is the value of \( f(x) \) when \( x = 0 \):
\[
f(0) = (0)^2 + 2(0) + 1 = 1
\]
Thus, the y-intercept is at \( (0, 1) \).

#### Maximum/Minimum Value:
Since the coefficient of \( x^2 \) is positive (\( a = 1 \)), the parabola opens upwards and has a **minimum value** at the vertex. The minimum value of \( f(x) \) is:
\[
f(-1) = 0
\]

#### Domain:
The domain of this quadratic function is all real numbers:
\[
\text{Domain: } (-\infty, \infty)
\]

#### Range:
Since the parabola opens upwards and the minimum value is \( 0 \), the range is:
\[
\text{Range: } [0, \infty)
\]

---

### Graphing the Parabolas:
To plot the graphs of these two functions, note:
1. \( f(x) = 3x^2 \) is a narrower parabola with its vertex at \( (0, 0) \).
2. \( f(x) = x^2 + 2x + 1 \) is a parabola that shifts left to \( (-1, 0) \).

I'll provide a graph if needed. Would you like me to generate the graphs now?

### Additional Questions:
1. How do you calculate the vertex for a more complex quadratic function?
2. Can you explain how the leading coefficient affects the shape of the parabola?
3. What are some real-life examples where quadratic functions are used?
4. How would the graph of \( f(x) = -x^2 \) differ from these examples?
5. Could you solve a quadratic equation using the quadratic formula?

### Tip:
Always check if a quadratic can be factored before using the quadratic formula; it often simplifies the solution.

Find the vertex, axis of symmetry, x-intercepts, y-intercept, value of the max/min, domain, and range of the following quadratics and then graph the parabola. 1. f(x) = 3x^2 2. f(x) = x^2 + 2x + 1

Explore how to solve quadratic equations, find the vertex, axis of symmetry, intercepts, and graph the parabolas for f(x) = 3x^2 and f(x) = x^2 + 2x + 1. Includes step-by-step solutions with important algebraic concepts and graphing instructions. Ideal for Grades 9-10.

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