Write the quadratic function in standard form and sketech its graph. Identify the vertex.
h(x)=(x-6)^2+3
You gave the equation in vertex form
so the vertex is easily read to be (6,3)
for standard form, just expand your function
h(x) = x^2 -12x + 36 + 3
= x^2 - 12x + 39
To write the quadratic function in standard form and sketch its graph, we need to expand and simplify the given equation.
The given quadratic function is h(x) = (x - 6)² + 3.
To expand this equation, we will apply the formula (a - b)² = a² - 2ab + b²:
h(x) = (x - 6)² + 3
= (x - 6)(x - 6) + 3
= x² - 12x + 36 + 3
= x² - 12x + 39.
Now that we have the quadratic function in standard form, which is ax² + bx + c, we can identify the coefficients:
a = 1 (the coefficient of x²),
b = -12 (the coefficient of x),
c = 39.
The vertex of a quadratic function can be found using the formula x = -b/(2a) and substituting it into the quadratic function to find the y-coordinate.
In our case, a = 1 and b = -12:
x = -(-12) / (2 * 1)
x = 12/2
x = 6.
Substituting x = 6 into the quadratic function:
h(6) = (6)² - 12(6) + 39
= 36 - 72 + 39
= 3.
Therefore, the vertex of the given quadratic function is (6, 3).
You can now sketch the graph of the quadratic function:
1. Plot the vertex at (6, 3).
2. Since the coefficient of x² is positive (a = 1), the parabola opens upwards.
3. Determine additional points by substituting values of x into the quadratic function and finding the corresponding y-values.
4. Connect the plotted points to form a smooth curve.
5. Include the x-axis and y-axis with appropriate scaling.
Note: The graph of the quadratic function will be a U-shaped curve, opening upwards, with the vertex at (6, 3).