Find an equation of quadratic function the graph passes through A(2,3)B(2,3)and C(1,0) and draw the parabola
Since points A and B have the same coordinates, it means they lie on the vertex of the parabola. Therefore, the axis of symmetry is x=2.
Let's write the equation of the parabola in vertex form:
f(x) = a(x - h)^2 + k
where (h,k) is the vertex. Since the axis of symmetry is x=2, we know that h=2. Also, since the parabola passes through C(1,0), we can plug in x=1 and y=0 to get an equation we can solve for a:
0 = a(1 - 2)^2 + 3
0 = a(1) + 3
a = -3
So the equation of the parabola is:
f(x) = -3(x - 2)^2 + 3
To draw the parabola, we can plot the vertex first (2,3), then use the shape of the parabola to plot a few more points. Since a is negative, the parabola opens downwards. Here's the graph:
To find the equation of a quadratic function that passes through points A(2,3), B(2,3), and C(1,0), we can use the standard form equation of a quadratic function: y = ax^2 + bx + c.
Step 1: Find the value of a:
Since the given points A(2,3) and B(2,3) have the same y-coordinate, we can conclude that the vertex of the parabola lies at x = 2. Therefore, x = 2 is the axis of symmetry. The equation of the axis of symmetry is given by x = -b / (2a). Substituting x = 2 into this equation, we get 2 = -b / (2a).
Step 2: Find the value of b:
Now we need another point, C(1,0), to find the value of b. Substituting x = 1 and y = 0 into the quadratic equation, we get 0 = a(1)^2 + b(1) + c. Since we already know the value of a from Step 1, we can substitute it into the equation: 0 = a + b + c.
Step 3: Find the value of c:
Using the equation 0 = a + b + c, and knowing that a = 2a (from Step 1), we can substitute this into the equation: 0 = 2a + b + c.
Now we have two equations:
2 = -b / (2a) and 0 = 2a + b + c.
To simplify these equations further, we can multiply both sides of the first equation by 2a and rearrange: 4a = -b.
Substituting this value (-b) into the second equation, we get 0 = 2a + 4a + c which simplifies to 0 = 6a + c.
Now we have two equations:
4a = -b and 0 = 6a + c.
Step 4: Substitute known values into one of the equations and solve for a:
We can substitute the equation 4a = -b into 0 = 6a + c to get 0 = 6a + c, which we can rearrange to c = -6a.
Substituting c = -6a into the equation 0 = 6a + c, we get 0 = 6a - 6a, which simplifies to 0 = 0.
Since the equation 0 = 0 is always true, it means that a can be any value. Let's choose a = 1 for simplicity.
Substituting a = 1 into the equations, we get:
4(1) = -b => -b = 4 => b = -4
c = -6(1) => c = -6
Step 5: Write the equation of the quadratic function:
Now we have the values of a = 1, b = -4, and c = -6.
The equation of the quadratic function that passes through the given points A(2,3), B(2,3), and C(1,0) is:
y = ax^2 + bx + c
=> y = 1x^2 - 4x - 6
=> y = x^2 - 4x - 6
To draw the parabola, you can plot additional points using the equation y = x^2 - 4x - 6 and then connect the points to create the graph.