find the vertex and the axis of symmetry andf(x)=x^2-10x+21`
graph the function
there are many ways to solve for the vertex. i think the easiest is to first find the derivative and solve for x:
f(x)=x^2-10x+21
f'(x) = 0 = 2x - 10
2x = 10
x = 5
now substitute this back to original:
f(5) = 5^2 - 10*5 + 21
f(5) = 25 - 50 + 21
f(5) = -4
thus vertex is at (5, -4).
and axis of symmetry is x = 5.
hope this helps~ :)
To find the vertex and axis of symmetry of the function f(x) = x^2 - 10x + 21, follow these steps:
Step 1: Identify the coefficients in the quadratic equation. In this case, the coefficient of x^2 is 1, the coefficient of x is -10, and the constant term is 21.
Step 2: Find the x-coordinate of the vertex using the formula: x = -b / (2a) for a quadratic equation in the form of f(x) = ax^2 + bx + c. Substituting the values into the formula, we get x = -(-10) / (2 * 1) = 10 / 2 = 5.
Step 3: Substitute the x-coordinate of the vertex back into the function to find the y-coordinate. Plugging x = 5 into the function, we have f(5) = (5)^2 - 10(5) + 21 = 25 - 50 + 21 = -4.
Therefore, the vertex is (5, -4).
Step 4: The axis of symmetry is a vertical line that passes through the vertex. In this case, the axis of symmetry is x = 5.
To graph the function, you can use the information obtained above and follow these steps:
Step 1: Plot the vertex (5, -4) on the coordinate plane.
Step 2: Find at least two other points on the graph by substituting different x-values into the function and calculating the corresponding y-values. For example, you can choose x = 0 and x = 10. Plugging x = 0 into the function, we have f(0) = (0)^2 - 10(0) + 21 = 0 - 0 + 21 = 21. So another point is (0, 21). Similarly, plugging x = 10 into the function, we have f(10) = (10)^2 - 10(10) + 21 = 100 - 100 + 21 = 21. So another point is (10, 21).
Step 3: Plot the additional points (0, 21) and (10, 21) on the graph.
Step 4: Draw a smooth curve that passes through the plotted points. The graph should be a parabola opening upward.
Step 5: Draw a dashed vertical line through the vertex (5, -4) to represent the axis of symmetry (x = 5).
Your graph of the function f(x) = x^2 - 10x + 21 should resemble a "U" shape opening upward with the vertex at (5, -4), and the axis of symmetry as a vertical line passing through x = 5.