sketch the graph of quadratic function. identify vertex and x-intercepts
f(x)=(x-3)squared-1
You know the value easily at x=3, you know that it is symettric about x=3, and an easy point to do in your head is x=8 and x=5
do those in your head, with the symettry about x=3, and you have the sketch.
Find a formula for the quadratic function whose graph has a y-intercept of y = 13 and zeros at x = 1, 6.
To sketch the graph of a quadratic function, such as f(x) = (x-3)^2 - 1, we can follow these steps:
Step 1: Identify the vertex
The vertex of a quadratic function in the form f(x) = a(x-h)^2 + k is given by the coordinates (h, k). In our case, the vertex is obtained by setting the expression inside the squared term equal to zero and solving for x:
x - 3 = 0
x = 3
So the vertex is located at (3, -1).
Step 2: Find the x-intercepts
The x-intercepts occur when the quadratic function equals zero. In other words, we need to solve the equation f(x) = 0. Let's set up the equation using our original function:
(x - 3)^2 - 1 = 0
(x - 3)^2 = 1
Taking the square root of both sides:
x - 3 = ± 1
Solving for x:
x = 3 + 1 = 4
x = 3 - 1 = 2
So the x-intercepts are located at (4, 0) and (2, 0).
Step 3: Sketch the graph
With the vertex and x-intercepts identified, we can now sketch the graph. The vertex is at (3, -1), which is the lowest point on the graph since the coefficient of the squared term (a in the general form) is positive. The x-intercepts are at (4, 0) and (2, 0).
To sketch the quadratic function, start by plotting the vertex at (3, -1). Next, plot the x-intercepts at (4, 0) and (2, 0). Since the function is symmetric because of the squared term, the graph will be a U-shaped curve opening upwards.
Label the vertex as (-1) and draw the curve from one x-intercept to the other, passing through the vertex. The final sketch of the graph should resemble a parabola with a minimum point at the vertex (3, -1).
Note: It is a good practice to plot a few additional points on either side of the x-intercepts to further understand the shape of the graph.