How do you graph the function h(x)=x2−3x+5*h*(*x*)=*x*2−3*x*+5 and identify key points?

To graph the function \( h(x) = x^2 - 3x + 5 \), you can follow these steps:

### 1. **Identify the form of the function**:
The given function \( h(x) = x^2 - 3x + 5 \) is a quadratic function of the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = -3 \), and \( c = 5 \). The graph of a quadratic function is a parabola.

### 2. **Find the vertex**:
The vertex of a parabola in the form \( ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:
\[
x_v = \frac{-b}{2a} = \frac{-(-3)}{2(1)} = \frac{3}{2} = 1.5
\]

Now, substitute \( x_v = 1.5 \) into the original equation to find the y-coordinate of the vertex:
\[
h(1.5) = (1.5)^2 - 3(1.5) + 5 = 2.25 - 4.5 + 5 = 2.75
\]
So, the vertex is \( (1.5, 2.75) \).

### 3. **Determine the axis of symmetry**:
The axis of symmetry is the vertical line that passes through the vertex. The equation of the axis of symmetry is:
\[
x = 1.5
\]

### 4. **Find the y-intercept**:
To find the y-intercept, set \( x = 0 \) in the function:
\[
h(0) = 0^2 - 3(0) + 5 = 5
\]
So, the y-intercept is \( (0, 5) \).

### 5. **Find additional points**:
You can find other points on the graph by selecting values of \( x \) around the vertex and calculating the corresponding \( y \)-values.

- For \( x = 0 \), \( h(0) = 5 \)
- For \( x = 1 \), \( h(1) = 1^2 - 3(1) + 5 = 3 \)
- For \( x = 2 \), \( h(2) = 2^2 - 3(2) + 5 = 3 \)

### 6. **Determine the direction of the parabola**:
Since the coefficient of \( x^2 \) (i.e., \( a = 1 \)) is positive, the parabola opens upwards.

### 7. **Plot the points and sketch the graph**:
Now that you have the vertex \( (1.5, 2.75) \), the y-intercept \( (0, 5) \), and additional points \( (1, 3) \) and \( (2, 3) \), you can plot these points and draw the parabola.

### Key Points:
- **Vertex**: \( (1.5, 2.75) \)
- **Y-intercept**: \( (0, 5) \)
- **Symmetry**: About the line \( x = 1.5 \)
- **Direction**: Opens upwards

Would you like more details on any part of this explanation?

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Here are 5 related questions to explore further:

1. How do you find the x-intercepts of a quadratic function?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you shift and scale a parabola by changing the coefficients of a quadratic equation?
4. How do you find the maximum or minimum value of a quadratic function?
5. How can you use factoring or the quadratic formula to solve \( x^2 - 3x + 5 = 0 \)?

**Tip**: If you're not sure where a quadratic opens up or down, always check the sign of the \( x^2 \) term—positive means upwards, negative means downwards.

How do you graph the function h(x) = x^2 − 3x + 5 and identify key points?

Learn how to graph the quadratic function h(x) = x^2 − 3x + 5 with step-by-step instructions. This guide covers how to find the vertex, axis of symmetry, y-intercept, and plot key points. Suitable for students in Grades 9-11.

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