f(x) = x^2 + 10x + 24
=(x^2 + 10x)+ 24
=(x^2 + 10x + 25 - 25) + 24
f(x) =(x+5)^2 - 1
Vertex: (-5, -1)
How do you sketch this? I know that in the graph one of the coord is (-5, -1) but how do I find the others in order to make a parabola? :|
Factorize to get the zeroes:
f(x)=x²+10x+24=(x+6)(x+4)
Therefore the zeroes at at x=-6, x=-4.
The y-intercept is y=24 (= constant term)
The coefficient of x² is positive, so the curve is concave upwards.
The vertex is the minimum of the parabola, and x=-5 passing through the vertex is the axis of symmetry.
That should give enough information to sketch the graph?
To sketch the graph of the function f(x) = (x + 5)^2 - 1, you can follow these steps:
1. Determine the vertex of the parabola: The vertex of a parabola in the form f(x) = (x - h)^2 + k is given by the coordinates (h, k). In this case, the vertex is (-5, -1), which you've already found.
2. Find the y-intercept: The y-intercept is the point where the graph intersects the y-axis. To find it, substitute x = 0 into the equation and solve for f(x). In this case, f(x) = (0 + 5)^2 - 1 = 25 - 1 = 24. So the y-intercept is (0, 24).
3. Find the x-intercepts (if any): The x-intercepts are the points where the graph intersects the x-axis. To find them, set f(x) = 0 and solve the equation. In this case, we have (x + 5)^2 - 1 = 0. Rewriting it as (x + 5)^2 = 1, take the square root of both sides to get x + 5 = ±1. Solving for x, we have x = -5 + 1 = -4 and x = -5 - 1 = -6. So the x-intercepts are (-4, 0) and (-6, 0).
4. Plot the vertex, y-intercept, and x-intercepts on a coordinate plane: Draw a set of axes and locate the vertex at (-5, -1), the y-intercept at (0, 24), and the x-intercepts at (-4, 0) and (-6, 0).
5. Sketch the parabolic curve: Since you know the vertex and a few other points on the graph, you can now sketch the curve. Remember that a parabola opens upward if the coefficient of the x^2 term is positive, as in this case, where it is 1. The curve should be a symmetric "U" shape with the vertex as its lowest point.
By following these steps, you should be able to sketch the graph of the function f(x) = (x + 5)^2 - 1, with the given coordinates and information.