In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-4) The graph passes through the point (1,4).

Verified step by step guidance

1. <insert step 1: Start with the vertex form of a parabola's equation, which is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.>
2. <insert step 2: Substitute the vertex (-3, -4) into the equation, replacing h with -3 and k with -4. This gives us y = a(x + 3)^2 - 4.>
3. <insert step 3: Use the point (1, 4) that the parabola passes through to find the value of a. Substitute x = 1 and y = 4 into the equation: 4 = a(1 + 3)^2 - 4.>
4. <insert step 4: Simplify the equation to solve for a. Start by calculating (1 + 3)^2, which is 16, and then solve 4 = 16a - 4.>
5. <insert step 5: Add 4 to both sides to get 8 = 16a, and then divide both sides by 16 to find the value of a.>

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is expressed as y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form is particularly useful for graphing and understanding the transformations of the parabola, as it directly shows the vertex's position and the direction of the opening based on the value of 'a'.

Finding 'a' using a Point

To determine the value of 'a' in the vertex form equation, you can substitute the coordinates of a known point on the parabola into the equation. This allows you to solve for 'a' by using the vertex coordinates and the coordinates of the point through which the parabola passes, ensuring the equation accurately represents the graph.

Graphing Parabolas

Graphing parabolas involves plotting the vertex and using the value of 'a' to determine the width and direction of the parabola's opening. Understanding the symmetry of parabolas and how they reflect across the axis of symmetry is crucial for accurately sketching the graph and identifying key features such as intercepts and the direction of the curve.