Question

identify the vertex of the parabola:
remember that the vertex is a point!

identify the y-intercept of the parabola:
remember that the y-intercept is a point!

identify the x-intercepts: and
remember that the x-intercepts represent points on the graph!

given the x-intercepts above, write an equation for the parabola in factored form:
\\(y = \\)
hint: think about the zero-product property.

write an equation for the axis of symmetry:

Explanation:

⚡ Using what you learned: Graphing Polynomial Functions

Step 1: Identify the vertex

The vertex is the lowest point (minimum) of this upward-opening parabola. Looking at the graph, the lowest point is located at:

$$ (-3, -1) $$

Step 2: Identify the y-intercept

The y-intercept is the point where the graph crosses the vertical y-axis. Looking at the y-axis, the curve crosses at:

$$ (0, 8) $$

Step 3: Identify the x-intercepts

The x-intercepts are the points where the graph crosses the horizontal x-axis. Looking at the x-axis, the curve crosses at:

$$ (-4, 0) \text{ and } (-2, 0) $$

Step 4: Write the equation in factored form

The factored form of a quadratic equation is:

$$ y = a(x - p)(x - q) $$

where \( p \) and \( q \) are the x-intercepts \( -4 \) and \( -2 \):

$$ y = a(x - (-4))(x - (-2)) $$

$$ y = a(x + 4)(x + 2) $$

To find the stretch factor \( a \), substitute the y-intercept \( (0, 8) \):

$$ 8 = a(0 + 4)(0 + 2) $$

$$ 8 = a(4)(2) $$

$$ 8 = 8a $$

$$ a = 1 $$

Thus, the equation in factored form is:

$$ y = (x + 4)(x + 2) $$

Step 5: Write the equation for the axis of symmetry

The axis of symmetry is the vertical line that passes through the x-coordinate of the vertex:

$$ x = -3 $$

Answer:

• Vertex: (-3, -1)
• y-intercept: (0, 8)
• x-intercepts: (-4, 0) and (-2, 0)
• Equation in factored form: y = (x + 4)(x + 2)
• Axis of symmetry: x = -3