For the following parabolas, fill in the table which follows. (12 Marks)
Parabola Graph
Vertex
Optimal Value
Axis of Symmetry
Zeroes
Direction of Opening
Y – intercept
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To fill in the table for the given parabolas, we need to analyze their characteristics based on the given information. Let's go step by step:
Step 1: Vertex
The vertex of a parabola can be determined from the given graph. It represents the "turning point" of the parabola, where it changes direction.
Step 2: Optimal Value
The optimal value of a parabola represents the maximum or minimum value achieved by the parabola.
Step 3: Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola and divides it into two equal halves.
Step 4: Zeroes
The zeroes of a parabola represent the x-values where the parabola crosses the x-axis or intersects with it. These are the points where the y-value (or the output) of the parabola is equal to zero.
Step 5: Direction of Opening
The direction of opening of a parabola can be determined by the coefficient in front of the x^2 term. If the coefficient is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards.
Step 6: Y-intercept
The y-intercept of a parabola is the point where the parabola crosses the y-axis, which occurs when x = 0.
Now, using these steps, we can fill in the table with the provided information for each parabola.
To fill in the table for the given parabolas, you need to understand the key properties of parabolas.
1. Vertex: The vertex is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards. To find the vertex, you can use the formula x = -b / (2a) for a quadratic equation in the form ax^2 + bx + c.
2. Optimal Value: The optimal value is the y-coordinate of the vertex. Once you have found the x-coordinate of the vertex, substitute it into the quadratic equation to find the y-coordinate.
3. Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry is x = the x-coordinate of the vertex.
4. Zeroes: The zeroes (also known as roots or x-intercepts) of a parabola are the points where the parabola intersects the x-axis. To find the zeroes, solve the quadratic equation for x = 0.
5. Direction of Opening: The direction of opening can be determined by looking at the coefficient of the x^2 term in the quadratic equation. If it is positive, the parabola opens upwards. If it is negative, the parabola opens downwards.
6. Y-intercept: The y-intercept is the point where the parabola intersects the y-axis. To find the y-intercept, substitute x = 0 into the quadratic equation.
By using these properties and formulas, you can fill in the table for the given parabolas.