determine the vertex for the function
y=2x^2+5x-3
for y = ax^2 + bx + c, the x value of the vertex is -b/(2a)
sub that x into your equation to find the y.
Another way is to complete the square.
so for yours, x = -5/4 etc.
a third way, if you know Calculus, is to find the derivative of y, set that equal to zero. Solving that for x gives you the x of the vertex, sub into the original equation to find the y.
To determine the vertex of the function y = 2x^2 + 5x - 3, we can use a few different methods.
Method 1: Using the formula for the x-value of the vertex
For a quadratic function in the form y = ax^2 + bx + c, the x-value of the vertex can be found using the formula x = -b / (2a).
In this case, a = 2 and b = 5. Plugging in these values, we can calculate the x-value of the vertex:
x = -5 / (2 * 2) = -5/4.
Now, to find the y-value of the vertex, we substitute the x-value back into the original equation:
y = 2(-5/4)^2 + 5(-5/4) - 3. Simplifying this expression will give us the y-value of the vertex.
Method 2: Completing the square
Another way to find the vertex is by completing the square. This involves rewriting the quadratic equation in a perfect square format. Here's how it's done:
Step 1: Group the x^2 and x terms together:
y = 2x^2 + 5x - 3
= 2(x^2 + 5/2x) - 3
Step 2: Add and subtract the square of half the coefficient of x from the expression inside the parentheses:
y = 2(x^2 + 5/2x + (5/4)^2 - (5/4)^2) - 3
= 2(x^2 + 5/2x + 25/16 - 25/16) - 3
= 2((x + 5/4)^2 - 25/16) - 3
= 2(x + 5/4)^2 - 2(25/16) - 3
= 2(x + 5/4)^2 - 25/8 - 3
= 2(x + 5/4)^2 - 25/8 - 24/8
= 2(x + 5/4)^2 - 49/8
Now, we can see that the vertex is at the point (-5/4, -49/8).
Method 3: Using calculus
If you are familiar with calculus, you can find the vertex by finding the derivative of the function y = 2x^2 + 5x - 3, setting it equal to zero, and solving for x. Then, substitute the x-value back into the original equation to find the y-value.
These methods provide different approaches to finding the vertex of the quadratic function. Choose the method that best suits your understanding and resources.