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.