Consider the quadratic function below and​ (a) find the​ vertex; (b) find the axis of​ symmetry; (c) determine whether there is a maximum or a minimum​ value, and find that​ value; and​ (d) graph the function. Equation: f(x)=x^2-10x+24

Let's solve the problem step by step, given the quadratic function:

\[
f(x) = x^2 - 10x + 24
\]

### (a) Finding the Vertex

The vertex of a quadratic function in the form \( f(x) = ax^2 + bx + c \) can be found using the formula:

\[
x = -\frac{b}{2a}
\]

Here, \( a = 1 \), \( b = -10 \), and \( c = 24 \). Plugging in these values:

\[
x = -\frac{-10}{2 \cdot 1} = \frac{10}{2} = 5
\]

To find the \( y \)-coordinate of the vertex, substitute \( x = 5 \) back into the function:

\[
f(5) = (5)^2 - 10(5) + 24 = 25 - 50 + 24 = -1
\]

Thus, the vertex is at \( (5, -1) \).

### (b) Finding the Axis of Symmetry

The axis of symmetry for a quadratic function is the vertical line that passes through the \( x \)-coordinate of the vertex. Therefore, the axis of symmetry is:

\[
x = 5
\]

### (c) Determining Maximum or Minimum Value

Since the coefficient of \( x^2 \) (i.e., \( a = 1 \)) is positive, the parabola opens upward, indicating that the vertex represents a minimum point.

The minimum value of the function is the \( y \)-coordinate of the vertex, which is:

\[
f(5) = -1
\]

### (d) Graphing the Function

To graph the function \( f(x) = x^2 - 10x + 24 \):

1. Plot the vertex at \( (5, -1) \).
2. Draw the axis of symmetry at \( x = 5 \).
3. Find the \( y \)-intercept by evaluating \( f(0) \):

   \[
   f(0) = 0^2 - 10(0) + 24 = 24
   \]

   So, the point \( (0, 24) \) is on the graph.

4. Find another point by choosing \( x = 10 \) (symmetric to \( x = 0 \) about \( x = 5 \)):

   \[
   f(10) = 10^2 - 10(10) + 24 = 100 - 100 + 24 = 24
   \]

   Thus, the point \( (10, 24) \) is also on the graph.

5. Draw a smooth curve passing through these points, forming a parabola opening upward.

Would you like me to provide a detailed graph or further explanation?

### Relative Questions:
1. How can the discriminant help determine the nature of the roots of this quadratic equation?
2. What is the significance of the \( y \)-intercept in the context of graphing a quadratic function?
3. How does changing the value of \( a \) in a quadratic function affect the graph's shape?
4. What happens to the graph of the quadratic function if you add a constant term to \( f(x) \)?
5. How do we solve a quadratic equation by completing the square?

### Tip:
For quadratic functions, the sign of \( a \) tells you whether the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)). This directly relates to whether the vertex represents a minimum or maximum.

Consider the quadratic function below and​ (a) find the​ vertex; (b) find the axis of​ symmetry; (c) determine whether there is a maximum or a minimum​ value, and find that​ value; and​ (d) graph the function. Equation: f(x) = x^2 - 10x + 24

Step-by-step solution for solving the quadratic function f(x) = x^2 - 10x + 24. Learn how to find the vertex, axis of symmetry, determine the minimum value, and graph the function. Suitable for high school students in Grades 9-11.

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