How do you sketch an equation like
x+4=-2(y-3)^2 ???
Make a table of (x,y) points and plot them. Let y assume values of -10 to 10 and calculate the corresponding x.
For x = -4, y = 3, for example
The curve will be a parabola on its side, opening to the left. x cannot exceed -4.
To sketch the equation x + 4 = -2(y - 3)², follow these steps:
Step 1: Rewrite the equation in a standard form for easier analysis. Expand the squared term by multiplying it with itself.
x + 4 = -2(y - 3)(y - 3)
Step 2: Simplify the equation further by distributing the -2 across the expression (y - 3)(y - 3):
x + 4 = -2(y² - 6y + 9)
Step 3: Continue simplification by distributing the -2 across the terms inside the parentheses:
x + 4 = -2y² + 12y - 18
Step 4: Set the equation equal to zero by moving all terms to one side:
x + 2y² - 12y + 22 = 0
Step 5: Analyze the equation to determine its shape and orientation:
The equation x + 2y² - 12y + 22 = 0 represents a quadratic equation, which has a certain general shape and orientation. The coefficient of the x term is positive (1), indicating the parabola will open to the right. The coefficient of the y² term is positive (2), indicating that the parabola will open upward.
Step 6: Plot the vertex:
To locate the vertex, use the formula -b/2a, where a and b are the coefficients of the y² and y terms, respectively. In this equation, a = 2 and b = -12. Therefore, the x-coordinate of the vertex is -(-12)/2(2), which simplifies to 3.
Step 7: Substitute the x-coordinate into the equation to find the y-coordinate of the vertex:
x + 2y² - 12y + 22 = 0
3 + 2y² - 12y + 22 = 0
2y² - 12y + 25 = 0
Step 8: Solve the quadratic equation to find the y-coordinate of the vertex. You can use the quadratic formula or factoring to find the y-coordinate. For this example, let's use the quadratic formula:
y = [-(-12) ± √((-12)² - 4(2)(25))] / (2*2)
y = [12 ± √(144 - 200)] / 4
y = [12 ± √(-56)] / 4
Since the equation inside the square root is negative, the equation has no real solutions. However, because the parabola opens upward, the vertex represents the minimum point of the parabola.
Step 9: Plot the vertex on the graph. In this case, the vertex is (3, y). Although we don't have the exact y-value, we can estimate its location based on the visual appearance of the equation.
Step 10: Draw the parabola:
Use the vertex and general shape of the quadratic equation to sketch a parabola that opens upward and has its vertex at (3, y).
To sketch an equation like x + 4 = -2(y - 3)^2, you can follow these steps:
1. Understand the equation: This equation represents a parabolic curve. It is in the form of vertex form, which is helpful in determining the shape and position of the parabola.
2. Analyze the vertex: The equation is written in the form (x - h) = a(y - k)^2, where the vertex is at the point (h, k). In this case, the vertex is the point (-4, 3). The vertex represents the highest or lowest point of the parabola.
3. Determine the axis of symmetry: The axis of symmetry is a vertical line that passes through the vertex. In this case, the axis of symmetry is x = -4.
4. Find the direction of the parabola: The coefficient of (y - k)^2, which is -2 in this equation, indicates the direction of the parabola. If the coefficient is positive, the parabola opens upwards. If it's negative, the parabola opens downwards. In this equation, the negative coefficient indicates that the parabola opens downwards.
5. Mark the vertex on the graph: Plot the point (-4, 3) on the coordinate plane. This represents the highest point or minimum point of the parabola.
6. Plot additional points: To sketch the parabola, pick a few x-values on each side of the vertex and substitute them into the equation to find their corresponding y-values. This will help you determine the shape of the parabola.
7. Sketch the parabola: Connect all the points you've plotted, considering the direction and shape of the parabola. Make sure the graph is smooth and symmetric about the axis of symmetry.
Following these steps should help you to sketch the given equation on a coordinate plane.