Graph the following functions showing the x-intercepts and the vertex.
y=x^2-10x+24
To graph the function y = x^2 - 10x + 24 and find its x-intercepts and vertex, we can start by finding the x-intercepts.
To find the x-intercepts, we set y = 0 and solve for x:
0 = x^2 - 10x + 24
This is a quadratic equation, so we can solve it by factoring or using the quadratic formula.
Factoring:
0 = (x - 4)(x - 6)
This means that either (x - 4) = 0 or (x - 6) = 0
Setting each factor equal to 0:
x - 4 = 0 --> x = 4
x - 6 = 0 --> x = 6
So the x-intercepts are x = 4 and x = 6.
Now let's find the vertex of the parabola using the formula x = -b/(2a), where a, b, and c are the coefficients of the quadratic equation.
In this case, a = 1, b = -10, and c = 24.
x = -(-10) / (2*1) = 10/2 = 5
Therefore, the x-coordinate of the vertex is x = 5.
To find the y-coordinate of the vertex, substitute x = 5 into the original equation:
y = (5)^2 - 10(5) + 24 = 25 - 50 + 24 = -1.
So the vertex is at (5, -1).
Plotting these points on the graph:
The x-intercepts are at (4, 0) and (6, 0).
The vertex is at (5, -1).
The graph of y = x^2 - 10x + 24 should look like a downward-opening parabola that passes through the x-intercepts at (4, 0) and (6, 0), and it has its vertex at (5, -1).
Here is the graph:
graph{y=x^2-10x+24 [-10, 10, -5, 10]}
To graph the function y = x^2 - 10x + 24 and find the x-intercepts and vertex, we can follow these steps:
Step 1: Find the x-intercepts (or the zeros) of the function by setting y = 0.
Setting y = 0:
0 = x^2 - 10x + 24
This quadratic equation can be factored as:
0 = (x - 4)(x - 6)
Setting each factor equal to 0:
x - 4 = 0 or x - 6 = 0
Solving for x:
x = 4 or x = 6
So the x-intercepts of the function are (4, 0) and (6, 0).
Step 2: Find the vertex of the function.
To find the vertex, we can use the formula: x = -b / 2a
In this case, the coefficient of x^2 is 1, and the coefficient of x is -10. Plugging these values into the formula:
x = -(-10) / (2*1)
x = 10 / 2 = 5
To find the corresponding y-coordinate of the vertex, we substitute this x-value back into the original equation:
y = (5)^2 - 10(5) + 24
y = 25 - 50 + 24
y = -1
So the vertex of the function is (5, -1).
Step 3: Graph the function.
The x-intercepts are (4, 0) and (6, 0), and the vertex is (5, -1).
Now we can plot these points on the Cartesian plane and sketch the graph of the function.