Question

fill in the info below, and graph on your own paper. then graph by clicking on the vertex, and then on another point close to the vertex. it is best to be able to click on an x or y intercept.

\\y = x^2 - 10x + 24\\

1. state whether the parabola opens up or down?
2. state the vertex as an ordered pair:
3. state the \\(y\\)-intercept as an ordered pair:
4. write the equation of the axis of symmetry:
5. click on the vertex and then another point of the parabola

Explanation:

🆕 New Concept Discovered: Graphing Quadratic Functions
Analyzing parabolas using key features

Step 1: Determine Direction of Opening

The quadratic equation is given in standard form:

$$ y = ax^2 + bx + c $$

$$ y = x^2 - 10x + 24 $$

Here, \( a = 1 \), \( b = -10 \), and \( c = 24 \).

Since \( a > 0 \), the parabola opens up.

Step 2: Find the Vertex

The \( x \)-coordinate of the vertex is found using:

$$ x = -\frac{b}{2a} $$

$$ x = -\frac{-10}{2(1)} = 5 $$

Substitute \( x = 5 \) back into the equation to find the \( y \)-coordinate:

$$ y = (5)^2 - 10(5) + 24 $$

$$ y = 25 - 50 + 24 = -1 $$

The vertex is \( (5, -1) \).

Step 3: Find the y-intercept

The \( y \)-intercept occurs where \( x = 0 \):

$$ y = (0)^2 - 10(0) + 24 = 24 $$

The \( y \)-intercept as an ordered pair is \( (0, 24) \).

Step 4: Find the Axis of Symmetry

The axis of symmetry is the vertical line passing through the \( x \)-coordinate of the vertex:

$$ x = 5 $$

Step 5: Identify Another Point on the Parabola

To graph the parabola, we can find another point near the vertex. Let's evaluate the function at \( x = 4 \):

$$ y = (4)^2 - 10(4) + 24 $$

$$ y = 16 - 40 + 24 = 0 $$

So, another point on the parabola is \( (4, 0) \) (which is also an \( x \)-intercept).

Answer:

1. Direction of opening: Up
2. Vertex: \( (5, -1) \)
3. y-intercept: \( (0, 24) \)
4. Equation of the Axis of Symmetry: \( x = 5 \)
5. Points to click for graphing: Vertex at \( (5, -1) \) and another point at \( (4, 0) \) (or \( (6, 0) \))