Graphing Quadratic Functions: Algebra 1 Math Worksheet – PDF Download

Problem 1: Name the coefficients

Write the values of \(a\), \(b\) and \(c\) for each function.

1. \(f(x)=3x^{2}-5x+2\)
2. \(g(x)=-x^{2}+4\)

Problem 2: Opens up or down

For each function, state whether the parabola opens up or down, whether the vertex is a maximum or a minimum, and the y-intercept.

1. \(f(x)=-2x^{2}+7x-4\)
2. \(h(x)=\frac{1}{2}x^{2}-3\)

Problem 3: Axis of symmetry

Find the equation of the axis of symmetry for each parabola.

1. \(y=x^{2}+8x+3\)
2. \(y=2x^{2}-12x+1\)
3. \(y=-x^{2}+5x\)

Problem 4: Vertex from standard form

Find the vertex of each parabola.

1. \(y=x^{2}-6x+5\)
2. \(y=-2x^{2}+8x-3\)

Problem 5: Read vertex form

Give the vertex, the axis of symmetry, and the maximum or minimum value of each function.

1. \(y=2(x-3)^{2}+1\)
2. \(y=-(x+2)^{2}+5\)

Problem 6: Zeros from factored form

The function \(y=(x+2)(x-6)\) is in factored form. Find its x-intercepts, its axis of symmetry and its vertex.

Problem 7: Complete a table

Complete the table for \(y=x^{2}-4x+1\). Then name the vertex and the axis of symmetry.

x	-1	0	1	2	3	4	5
y	?	?	?	?	?	?	?

Problem 8: Read a graph

Use the graph to name the vertex, the axis of symmetry, the x-intercepts, the y-intercept, and the range of the function.

Problem 9: Effect of a

The equations \(y=x^{2}\), \(y=2x^{2}\) and \(y=\frac{1}{2}x^{2}\) are graphed below. Match each equation to its curve. Then describe how the graph of \(y=-2x^{2}\) differs from curve B.

Problem 10: Intercepts from standard form

Find the x-intercepts and the y-intercept of \(y=x^{2}-5x+6\).

Problem 11: Maximum or minimum value

Does \(f(x)=-x^{2}+6x-5\) have a maximum or a minimum? Find the value and where it occurs. Then find the zeros.

Problem 12: Factored to standard form

Write \(y=(x-3)(x+5)\) in standard form. Then find its vertex.

Problem 13: Complete the square

Rewrite \(y=x^{2}+8x+7\) in vertex form by completing the square. State the vertex.

Problem 14: No x-intercepts

Complete the square for \(y=x^{2}-10x+30\). Find the minimum value and explain why the graph has no x-intercepts.

Problem 15: Leading coefficient 2

Write \(y=2x^{2}-12x+10\) in vertex form. Then find the zeros.

Problem 16: Increasing and decreasing

The graph of \(f(x)=-x^{2}+4x\) is shown. State the interval where \(f\) is increasing, the interval where it is decreasing, and the interval where \(f(x) > 0\).

Problem 17: Equation from a graph

The graph shows a parabola with vertex \((1, 4)\) that passes through \((3, 0)\). Write its equation in vertex form, then in standard form.

Problem 18: Compare two functions

Function \(f\) is \(f(x)=x^{2}-6x+11\). Function \(g\) is given by the table. Which function has the lower minimum value?

x	0	1	2	3	4
g(x)	5	2	1	2	5

Problem 19: Three forms, one parabola

Show that \(y=(x-3)^{2}-4\) and \(y=(x-1)(x-5)\) are both equal to \(y=x^{2}-6x+5\). Then tell which form shows the vertex, which shows the zeros, and which shows the y-intercept.

Problem 20: Equation from zeros and a point

A parabola has x-intercepts at 2 and 8 and passes through \((5, -9)\). Write its equation in factored form and in standard form. Find the vertex.

Problem 21: Thrown ball

Maya throws a ball upward from a balcony. Its height in feet after \(t\) seconds is \(h(t)=-16t^{2}+32t+6\).

1. How high is the ball at \(t=0\)?
2. When does it reach its highest point, and how high is that?
3. At what other time is the ball again at the height from part (a)?

Problem 22: Garden fence

Diego has 40 feet of fence for a rectangular garden. One side is a wall, so only three sides need fence. Let \(x\) be the length of each of the two sides that touch the wall.

1. Write a function \(A(x)\) for the area.
2. Find the value of \(x\) that gives the largest area.
3. Find that area and the dimensions of the garden.

Problem 23: T-shirt sale

The school store sells 100 T-shirts at \(\$12\) each. For every \(\$1\) the price goes up, the store sells 5 fewer shirts. With \(x\) dollars of increase, the revenue is \(R(x)=(12+x)(100-5x)\).

1. Write \(R(x)\) in standard form.
2. What price gives the greatest revenue?
3. Find the greatest revenue and the number of shirts sold.

Problem 24: Fountain arc

A stream of water from a fountain follows \(y=-\frac{1}{4}x^{2}+2x\), where \(x\) is the horizontal distance and \(y\) is the height, both in feet.

1. Find the greatest height of the water.
2. How far from the nozzle does the water land?
3. Find the height at \(x=2\), and another distance at which the water has that same height.

See every chapter on the Algebra 1 math page.