To find the equation of a quadratic function given its graph, we need to identify the key points on the graph. From the given graph, we can see that the quadratic function intersects the y-axis at the point (0, 4). This gives us the value of the y-intercept, which is 4.
Next, we can identify the x-intercepts of the graph. From the graph, we can see that the quadratic function intersects the x-axis at the points (-3, 0) and (5, 0). These points give us the roots or solutions of the quadratic equation.
Now, let's use the standard form of a quadratic equation:
\[y = ax^2 + bx + c\]
We know that the y-intercept is 4, so when x = 0, y = 4. Substituting these values into the equation, we get:
\[4 = a(0)^2 + b(0) + c\]
Simplifying, we find that c = 4.
Next, we can use the x-intercepts (-3, 0) and (5, 0) to find the values of a and b. When x = -3, y = 0. Substituting these values into the equation, we get:
\[0 = a(-3)^2 + b(-3) + 4\]
Simplifying, we get:
\[9a - 3b + 4 = 0\] ---> (Equation 1)
Similarly, when x = 5, y = 0. Substituting these values into the equation, we get:
\[0 = a(5)^2 + b(5) + 4\]
Simplifying, we get:
\[25a + 5b + 4 = 0\] ---> (Equation 2)
We now have two equations (Equation 1 and Equation 2) with two unknowns (a and b). Solving these equations simultaneously will give us the values of a and b, which we can then substitute back into the equation \(y = ax^2 + bx + c\) to obtain the final equation of the quadratic function.
Solving Equations 1 and 2, we find that a = -1 and b = 2.
Substituting these values into the equation \(y = ax^2 + bx + c\), we get:
\[y = -x^2 + 2x + 4\]
Therefore, the equation of the quadratic function is \[y = -x^2 + 2x + 4\].
Note: The correct answer option is the equation \(y = -x^2 + 2x + 4\).
Answers & Comments
Answer:
To find the equation of a quadratic function given its graph, we need to identify the key points on the graph. From the given graph, we can see that the quadratic function intersects the y-axis at the point (0, 4). This gives us the value of the y-intercept, which is 4.
Next, we can identify the x-intercepts of the graph. From the graph, we can see that the quadratic function intersects the x-axis at the points (-3, 0) and (5, 0). These points give us the roots or solutions of the quadratic equation.
Now, let's use the standard form of a quadratic equation:
\[y = ax^2 + bx + c\]
We know that the y-intercept is 4, so when x = 0, y = 4. Substituting these values into the equation, we get:
\[4 = a(0)^2 + b(0) + c\]
Simplifying, we find that c = 4.
Next, we can use the x-intercepts (-3, 0) and (5, 0) to find the values of a and b. When x = -3, y = 0. Substituting these values into the equation, we get:
\[0 = a(-3)^2 + b(-3) + 4\]
Simplifying, we get:
\[9a - 3b + 4 = 0\] ---> (Equation 1)
Similarly, when x = 5, y = 0. Substituting these values into the equation, we get:
\[0 = a(5)^2 + b(5) + 4\]
Simplifying, we get:
\[25a + 5b + 4 = 0\] ---> (Equation 2)
We now have two equations (Equation 1 and Equation 2) with two unknowns (a and b). Solving these equations simultaneously will give us the values of a and b, which we can then substitute back into the equation \(y = ax^2 + bx + c\) to obtain the final equation of the quadratic function.
Solving Equations 1 and 2, we find that a = -1 and b = 2.
Substituting these values into the equation \(y = ax^2 + bx + c\), we get:
\[y = -x^2 + 2x + 4\]
Therefore, the equation of the quadratic function is \[y = -x^2 + 2x + 4\].
Note: The correct answer option is the equation \(y = -x^2 + 2x + 4\).